3.5.19 \(\int \frac {(a+b x^2)^2}{\sqrt {x} (c+d x^2)} \, dx\) [419]

Optimal. Leaf size=266 \[ -\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}-\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}} \]

[Out]

2/5*b^2*x^(5/2)/d-1/2*(-a*d+b*c)^2*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(9/4)*2^(1/2)+1/2*(-a*d
+b*c)^2*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(9/4)*2^(1/2)-1/4*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2
)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(3/4)/d^(9/4)*2^(1/2)+1/4*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/
4)*2^(1/2)*x^(1/2))/c^(3/4)/d^(9/4)*2^(1/2)-2*b*(-2*a*d+b*c)*x^(1/2)/d^2

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Rubi [A]
time = 0.15, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {472, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {(b c-a d)^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}-\frac {2 b \sqrt {x} (b c-2 a d)}{d^2}+\frac {2 b^2 x^{5/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)),x]

[Out]

(-2*b*(b*c - 2*a*d)*Sqrt[x])/d^2 + (2*b^2*x^(5/2))/(5*d) - ((b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(Sqrt[2]*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*
c^(3/4)*d^(9/4)) - ((b*c - a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(3/
4)*d^(9/4)) + ((b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(3/4)*d^
(9/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )} \, dx &=\int \left (-\frac {b (b c-2 a d)}{d^2 \sqrt {x}}+\frac {b^2 x^{3/2}}{d}+\frac {b^2 c^2-2 a b c d+a^2 d^2}{d^2 \sqrt {x} \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{d^2}\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}+\frac {\left (2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {c} d^2}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {c} d^2}\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {c} d^{5/2}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {c} d^{5/2}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}\\ &=-\frac {2 b (b c-2 a d) \sqrt {x}}{d^2}+\frac {2 b^2 x^{5/2}}{5 d}-\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} d^{9/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{3/4} d^{9/4}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 155, normalized size = 0.58 \begin {gather*} \frac {4 b \sqrt [4]{d} \sqrt {x} \left (-5 b c+10 a d+b d x^2\right )-\frac {5 \sqrt {2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4}}+\frac {5 \sqrt {2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4}}}{10 d^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(Sqrt[x]*(c + d*x^2)),x]

[Out]

(4*b*d^(1/4)*Sqrt[x]*(-5*b*c + 10*a*d + b*d*x^2) - (5*Sqrt[2]*(b*c - a*d)^2*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt
[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/c^(3/4) + (5*Sqrt[2]*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(S
qrt[c] + Sqrt[d]*x)])/c^(3/4))/(10*d^(9/4))

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Maple [A]
time = 0.10, size = 159, normalized size = 0.60

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-b c \sqrt {x}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} c}\)	\(159\)
default	\(\frac {2 b \left (\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-b c \sqrt {x}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} c}\)	\(159\)
risch	\(\frac {2 \left (b d \,x^{2}+10 a d -5 b c \right ) b \sqrt {x}}{5 d^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} c \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 d^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{2 c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} c \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2}}{2 d^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{4 c}-\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{2 d}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} c \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2}}{4 d^{2}}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/(d*x^2+c)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*b/d^2*(1/5*b*d*x^(5/2)+2*a*d*x^(1/2)-b*c*x^(1/2))+1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^2*(c/d)^(1/4)/c*2^(1/2)*
(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/
(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1))

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Maxima [A]
time = 0.50, size = 291, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (b^{2} d x^{\frac {5}{2}} - 5 \, {\left (b^{2} c - 2 \, a b d\right )} \sqrt {x}\right )}}{5 \, d^{2}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{4 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)/x^(1/2),x, algorithm="maxima")

[Out]

2/5*(b^2*d*x^(5/2) - 5*(b^2*c - 2*a*b*d)*sqrt(x))/d^2 + 1/4*(2*sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(
1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d)
)) + 2*sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x
))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(sqrt(2
)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*l
og(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/d^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1245 vs. \(2 (193) = 386\).
time = 0.52, size = 1245, normalized size = 4.68 \begin {gather*} \frac {20 \, d^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{2} d^{4} \sqrt {-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x} c^{2} d^{7} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {3}{4}} - {\left (b^{2} c^{4} d^{7} - 2 \, a b c^{3} d^{8} + a^{2} c^{2} d^{9}\right )} \sqrt {x} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {3}{4}}}{b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}\right ) + 5 \, d^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {1}{4}} \log \left (c d^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 5 \, d^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {1}{4}} \log \left (-c d^{2} \left (-\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{9}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) + 4 \, {\left (b^{2} d x^{2} - 5 \, b^{2} c + 10 \, a b d\right )} \sqrt {x}}{10 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)/x^(1/2),x, algorithm="fricas")

[Out]

1/10*(20*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^
5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*arctan((sqrt(c^2*d^4*sqrt(-(b^8
*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*
a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9)) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b
*c*d^3 + a^4*d^4)*x)*c^2*d^7*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4
*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(3/4) - (b^2*c^4*d^7
- 2*a*b*c^3*d^8 + a^2*c^2*d^9)*sqrt(x)*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 +
70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(3/4))/(b^8
*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*
a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)) + 5*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b
^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9)
)^(1/4)*log(c*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 -
56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a
^2*d^2)*sqrt(x)) - 5*d^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4
*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4)*log(-c*d^2*(-(b^8*c
^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^
6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^9))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 4*(b^2*
d*x^2 - 5*b^2*c + 10*a*b*d)*sqrt(x))/d^2

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Sympy [A]
time = 4.69, size = 423, normalized size = 1.59 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}}{d} & \text {for}\: c = 0 \\\frac {2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}}{c} & \text {for}\: d = 0 \\- \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c} + \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c} + \frac {4 a b \sqrt {x}}{d} + \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d} - \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d} - \frac {2 a b \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d} - \frac {2 b^{2} c \sqrt {x}}{d^{2}} - \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {b^{2} c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {2 b^{2} x^{\frac {5}{2}}}{5 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/(d*x**2+c)/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5), Eq(c, 0) & Eq(d, 0)), ((-2*a**2/(3*
x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5)/d, Eq(c, 0)), ((2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(
9/2)/9)/c, Eq(d, 0)), (-a**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*c) + a**2*(-c/d)**(1/4)*log(sqrt(x)
+ (-c/d)**(1/4))/(2*c) + a**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/c + 4*a*b*sqrt(x)/d + a*b*(-c/d)**(1/
4)*log(sqrt(x) - (-c/d)**(1/4))/d - a*b*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/d - 2*a*b*(-c/d)**(1/4)*ata
n(sqrt(x)/(-c/d)**(1/4))/d - 2*b**2*c*sqrt(x)/d**2 - b**2*c*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*d**2
) + b**2*c*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d**2) + b**2*c*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/
4))/d**2 + 2*b**2*x**(5/2)/(5*d), True))

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Giac [A]
time = 1.24, size = 360, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{3}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{3}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{3}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{3}} + \frac {2 \, {\left (b^{2} d^{4} x^{\frac {5}{2}} - 5 \, b^{2} c d^{3} \sqrt {x} + 10 \, a b d^{4} \sqrt {x}\right )}}{5 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)/x^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt
(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c*d^3) + 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c
*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c*d^3) + 1/4*s
qrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/
4) + x + sqrt(c/d))/(c*d^3) - 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2
*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^3) + 2/5*(b^2*d^4*x^(5/2) - 5*b^2*c*d^3*sqrt(x) +
10*a*b*d^4*sqrt(x))/d^5

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Mupad [B]
time = 0.20, size = 1107, normalized size = 4.16 \begin {gather*} \frac {2\,b^2\,x^{5/2}}{5\,d}-\sqrt {x}\,\left (\frac {2\,b^2\,c}{d^2}-\frac {4\,a\,b}{d}\right )+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{3/4}\,d^{9/4}}+\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{3/4}\,d^{9/4}}}{\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{3/4}\,d^{9/4}}-\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{3/4}\,d^{9/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{3/4}\,d^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{3/4}\,d^{9/4}}+\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{3/4}\,d^{9/4}}}{\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{3/4}\,d^{9/4}}-\frac {\left (\frac {8\,\sqrt {x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (16\,a^2\,c\,d^3-32\,a\,b\,c^2\,d^2+16\,b^2\,c^3\,d\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{3/4}\,d^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{3/4}\,d^{9/4}}}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{3/4}\,d^{9/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/(x^(1/2)*(c + d*x^2)),x)

[Out]

(2*b^2*x^(5/2))/(5*d) - x^(1/2)*((2*b^2*c)/d^2 - (4*a*b)/d) + (atan(((((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b
^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2)
)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)) + (((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2
*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/
(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)))/((((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c
^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/(2
*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4)) - (((8*x^(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^
2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a*d - b*c)^2*(16*a^2*c*d^3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2))/(2*(-c)
^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4))))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)) + (atan(((((8*x^
(1/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^
3 + 16*b^2*c^3*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4)) + (((8*x^(1
/2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a*d - b*c)^2*(16*a^2*c*d^3
+ 16*b^2*c^3*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2)/((-c)^(3/4)*d^(9/4)))/((((8*x^(1/2
)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d - ((a*d - b*c)^2*(16*a^2*c*d^3 +
16*b^2*c^3*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4)) - (((8*x^(1/
2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/d + ((a*d - b*c)^2*(16*a^2*c*d^3 +
16*b^2*c^3*d - 32*a*b*c^2*d^2)*1i)/(2*(-c)^(3/4)*d^(9/4)))*(a*d - b*c)^2*1i)/((-c)^(3/4)*d^(9/4))))*(a*d - b*
c)^2)/((-c)^(3/4)*d^(9/4))

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