\(\int \frac {\sqrt {x} (c+d x^2)^3}{a+b x^2} \, dx\) [443]

Optimal result

Integrand size = 24, antiderivative size = 306 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}} \]

[Out]

2/3*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^(3/2)/b^3+2/7*d^2*(-a*d+3*b*c)*x^(7/2)/b^2+2/11*d^3*x^(11/2)/b-1/2*(-a*d
+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/b^(15/4)*2^(1/2)+1/2*(-a*d+b*c)^3*arctan(1+b^(1/4)*2
^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/b^(15/4)*2^(1/2)+1/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)
*x^(1/2))/a^(1/4)/b^(15/4)*2^(1/2)-1/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1
/4)/b^(15/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {472, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b} \]

[In]

Int[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11
/2))/(11*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c
- a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4))

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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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*} \text {integral}& = \int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {d^2 (3 b c-a d) x^{5/2}}{b^2}+\frac {d^3 x^{9/2}}{b}+\frac {\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) \sqrt {x}}{b^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^3} \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{7/2}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{7/2}} \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^4}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^4}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}} \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}} \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d x^{3/2} \left (77 a^2 d^2-33 a b d \left (7 c+d x^2\right )+3 b^2 \left (77 c^2+33 c d x^2+7 d^2 x^4\right )\right )}{231 b^3}+\frac {(-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}} \]

[In]

Integrate[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*d*x^(3/2)*(77*a^2*d^2 - 33*a*b*d*(7*c + d*x^2) + 3*b^2*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4)))/(231*b^3) + ((-(
b*c) + a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((
-(b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(1/4)*b^(15/4))

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {2 \left (21 b^{2} d^{2} x^{4}-33 x^{2} a b \,d^{2}+99 x^{2} b^{2} c d +77 a^{2} d^{2}-231 a b c d +231 b^{2} c^{2}\right ) d \,x^{\frac {3}{2}}}{231 b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(206\)
derivativedivides	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(208\)
default	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(208\)

[In]

int((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

2/231*(21*b^2*d^2*x^4-33*a*b*d^2*x^2+99*b^2*c*d*x^2+77*a^2*d^2-231*a*b*c*d+231*b^2*c^2)*d*x^(3/2)/b^3-1/4*(a^3
*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^4/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/
2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1992, normalized size of antiderivative = 6.51 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\text {Too large to display} \]

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/462*(231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8
*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3
*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(a*b^11*(-(b^12*c^12 - 12*a*b^1
1*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^
6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b
*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 12
6*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*s
qrt(x)) - 231*I*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8
*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3
*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(I*a*b^11*(-(b^12*c^12 - 12
*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*
a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*
a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^
3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*
d^9)*sqrt(x)) + 231*I*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a
^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a
^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(-I*a*b^11*(-(b^12*c^
12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5
+ 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^1
0 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6
*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8
- a^9*d^9)*sqrt(x)) - 231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(-a*b^11*(-(b^12
*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*
d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*
d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*
b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*
d^8 - a^9*d^9)*sqrt(x)) - 4*(21*b^2*d^3*x^5 + 33*(3*b^2*c*d^2 - a*b*d^3)*x^3 + 77*(3*b^2*c^2*d - 3*a*b*c*d^2 +
a^2*d^3)*x)*sqrt(x))/b^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (291) = 582\).

Time = 17.09 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}}{b} & \text {for}\: a = 0 \\\frac {\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}}{a} & \text {for}\: b = 0 \\\frac {2 a^{2} d^{3} x^{\frac {3}{2}}}{3 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} - \frac {2 a c d^{2} x^{\frac {3}{2}}}{b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {2 a d^{3} x^{\frac {7}{2}}}{7 b^{2}} + \frac {2 c^{2} d x^{\frac {3}{2}}}{b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7 b} + \frac {2 d^{3} x^{\frac {11}{2}}}{11 b} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x**2+c)**3*x**(1/2)/(b*x**2+a),x)

[Out]

Piecewise((zoo*(-2*c**3/sqrt(x) + 2*c**2*d*x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11), Eq(a, 0) & E
q(b, 0)), ((-2*c**3/sqrt(x) + 2*c**2*d*x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11)/b, Eq(a, 0)), ((2
*c**3*x**(3/2)/3 + 6*c**2*d*x**(7/2)/7 + 6*c*d**2*x**(11/2)/11 + 2*d**3*x**(15/2)/15)/a, Eq(b, 0)), (2*a**2*d*
*3*x**(3/2)/(3*b**3) + a**2*d**3*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**3) - a**2*d**3*(-a/b)**(3/4)
*log(sqrt(x) + (-a/b)**(1/4))/(2*b**3) + a**2*d**3*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**3 - 2*a*c*d**2
*x**(3/2)/b**2 - 3*a*c*d**2*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**2) + 3*a*c*d**2*(-a/b)**(3/4)*log
(sqrt(x) + (-a/b)**(1/4))/(2*b**2) - 3*a*c*d**2*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 - 2*a*d**3*x**(
7/2)/(7*b**2) + 2*c**2*d*x**(3/2)/b + 3*c**2*d*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - 3*c**2*d*(-a
/b)**(3/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) + 3*c**2*d*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 6*c*d**
2*x**(7/2)/(7*b) + 2*d**3*x**(11/2)/(11*b) - c**3*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + c**3*(-a/
b)**(3/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a) - c**3*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/a, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{3}} + \frac {2 \, {\left (21 \, b^{2} d^{3} x^{\frac {11}{2}} + 33 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{231 \, b^{3}} \]

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

1/4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4)
+ 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(
sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*
log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^
(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/b^3 + 2/231*(21*b^2*d^3*x^(11/2) + 33*(3*b^2*c*d^2 - a
*b*d^3)*x^(7/2) + 77*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^(3/2))/b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (227) = 454\).

Time = 0.30 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{6}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{6}} + \frac {2 \, {\left (21 \, b^{10} d^{3} x^{\frac {11}{2}} + 99 \, b^{10} c d^{2} x^{\frac {7}{2}} - 33 \, a b^{9} d^{3} x^{\frac {7}{2}} + 231 \, b^{10} c^{2} d x^{\frac {3}{2}} - 231 \, a b^{9} c d^{2} x^{\frac {3}{2}} + 77 \, a^{2} b^{8} d^{3} x^{\frac {3}{2}}\right )}}{231 \, b^{11}} \]

[In]

integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)
*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^6) + 1/2*sqrt(2)*((a*b^3)^(3/
4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sq
rt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^6) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^
(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x +
sqrt(a/b))/(a*b^6) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b
*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^6) + 2/231*(21*b^10*d^3
*x^(11/2) + 99*b^10*c*d^2*x^(7/2) - 33*a*b^9*d^3*x^(7/2) + 231*b^10*c^2*d*x^(3/2) - 231*a*b^9*c*d^2*x^(3/2) +
77*a^2*b^8*d^3*x^(3/2))/b^11

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{3/2}\,\left (\frac {2\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{3\,b}\right )-x^{7/2}\,\left (\frac {2\,a\,d^3}{7\,b^2}-\frac {6\,c\,d^2}{7\,b}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,b^{15/4}} \]

[In]

int((x^(1/2)*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^(3/2)*((2*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(3*b)) - x^(7/2)*((2*a*d^3)/(7*b^2) - (6*c*d^2)/(7*b)
) + (2*d^3*x^(11/2))/(11*b) - (atan((b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15
*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5))/((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9
+ 9*a^2*b^8*c^8*d - 36*a^3*b^7*c^7*d^2 + 84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 126*a^6*b^4*c^4*d^5 - 84*
a^7*b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a^9*b*c*d^8)))*(a*d - b*c)^3)/((-a)^(1/4)*b^(15/4)) - (atan((b^(1/4)*
x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^
5*b^2*c^2*d^4 - 6*a^6*b*c*d^5)*1i)/((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9 + 9*a^2*b^8*c^8*d - 36*a^3*b^7*c^7*d^2 +
84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 126*a^6*b^4*c^4*d^5 - 84*a^7*b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a
^9*b*c*d^8)))*(a*d - b*c)^3*1i)/((-a)^(1/4)*b^(15/4))