\(\int \frac {(c+d x^2)^3}{x^{3/2} (a+b x^2)} \, dx\) [445]

Optimal result

Integrand size = 24, antiderivative size = 284 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 b}+\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} b^{11/4}}-\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} b^{11/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} b^{11/4}} \]

[Out]

2/3*d^2*(-a*d+3*b*c)*x^(3/2)/b^2+2/7*d^3*x^(7/2)/b+1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/
a^(5/4)/b^(11/4)*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^(5/4)/b^(11/4)*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^(5/4)/b^(11/4)*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^(5/4)/b^(11/4)*2^(1/2)-2*c^3/a/x^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 284, 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 = {477, 472, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{5/4} b^{11/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{5/4} b^{11/4}}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} b^{11/4}}+\frac {2 d^2 x^{3/2} (3 b c-a d)}{3 b^2}-\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^3 x^{7/2}}{7 b} \]

[In]

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

[Out]

(-2*c^3)/(a*Sqrt[x]) + (2*d^2*(3*b*c - a*d)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(7*b) + ((b*c - a*d)^3*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*b^(11/4)) - ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*b^(11/4)) - ((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^(5/4)*b^(11/4)) + ((b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4))

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 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 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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}& = 2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {c^3}{a x^2}+\frac {d^2 (3 b c-a d) x^2}{b^2}+\frac {d^3 x^6}{b}+\frac {(-b c+a d)^3 x^2}{a b^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 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 )}{a b^2} \\ & = -\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 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 )}{a b^{5/2}}-\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 )}{a b^{5/2}} \\ & = -\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 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 a b^3}-\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 a b^3}-\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} a^{5/4} b^{11/4}}-\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} a^{5/4} b^{11/4}} \\ & = -\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 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} a^{5/4} b^{11/4}}+\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} a^{5/4} b^{11/4}}-\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} a^{5/4} b^{11/4}}+\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} a^{5/4} b^{11/4}} \\ & = -\frac {2 c^3}{a \sqrt {x}}+\frac {2 d^2 (3 b c-a d) x^{3/2}}{3 b^2}+\frac {2 d^3 x^{7/2}}{7 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} a^{5/4} b^{11/4}}-\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} b^{11/4}}-\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} a^{5/4} b^{11/4}}+\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} a^{5/4} b^{11/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=\frac {\frac {4 \sqrt [4]{a} b^{3/4} \left (-21 b^2 c^3-7 a^2 d^3 x^2+3 a b d^2 x^2 \left (7 c+d x^2\right )\right )}{\sqrt {x}}+21 \sqrt {2} (b c-a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{42 a^{5/4} b^{11/4}} \]

[In]

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

[Out]

((4*a^(1/4)*b^(3/4)*(-21*b^2*c^3 - 7*a^2*d^3*x^2 + 3*a*b*d^2*x^2*(7*c + d*x^2)))/Sqrt[x] + 21*Sqrt[2]*(b*c - a
*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 21*Sqrt[2]*(b*c - a*d)^3*ArcTanh[(Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(42*a^(5/4)*b^(11/4))

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (a d -3 b c \right ) x^{\frac {3}{2}}}{3}\right )}{b^{2}}-\frac {2 c^{3}}{a \sqrt {x}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \,b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(186\)
default	\(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (a d -3 b c \right ) x^{\frac {3}{2}}}{3}\right )}{b^{2}}-\frac {2 c^{3}}{a \sqrt {x}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \,b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(186\)
risch	\(-\frac {2 \left (-3 a b \,d^{3} x^{4}+7 a^{2} d^{3} x^{2}-21 a b c \,d^{2} x^{2}+21 b^{2} c^{3}\right )}{21 a \sqrt {x}\, b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \,b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(198\)

[In]

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

[Out]

-2*d^2/b^2*(-1/7*b*d*x^(7/2)+1/3*(a*d-3*b*c)*x^(3/2))-2*c^3/a/x^(1/2)+1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d
-b^3*c^3)/a/b^3/(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 = 0.29 (sec) , antiderivative size = 1993, normalized size of antiderivative = 7.02 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/42*(21*a*b^2*x*(-(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^5*b^11))^(1/4)*log(a^4*b^8*(-(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^5*b^11))^(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)) - 21*I*a*b^2*x*(-(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 + 49
5*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 - 22
0*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^5*b^11))^(1/4)*log(I*a^4*b^8*(-(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^5*b^11))^(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)) + 21*I*a*b^2*x*(-(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^5*b^11))^(1/4)*log(-
I*a^4*b^8*(-(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^5*b^11))^(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)) - 21*a*b^2*x*(-(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^5*b^11)
)^(1/4)*log(-a^4*b^8*(-(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^5*b^11))^(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 - 3
6*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 4*(3*a*b*d^3*x^4 - 21*b^2*c^3 + 7*(3*a*b*c*d^2 - a^2*d
^3)*x^2)*sqrt(x))/(a*b^2*x)

Sympy [A] (verification not implemented)

Time = 42.37 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.80 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=c^{3} \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a \sqrt [4]{- \frac {a}{b}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 c^{2} d \left (\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: a = 0 \\\frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b \sqrt [4]{- \frac {a}{b}}} - \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b \sqrt [4]{- \frac {a}{b}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + 3 c d^{2} \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + d^{3} \left (\begin {cases} \tilde {\infty } x^{\frac {7}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 b} & \text {for}\: a = 0 \\\frac {a^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3} \sqrt [4]{- \frac {a}{b}}} - \frac {a^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3} \sqrt [4]{- \frac {a}{b}}} + \frac {a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3} \sqrt [4]{- \frac {a}{b}}} - \frac {2 a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

c**3*Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b*x**(5/2)), Eq(a, 0)), (-2/(a*sqrt(x)), Eq(b, 0)),
(-log(sqrt(x) - (-a/b)**(1/4))/(2*a*(-a/b)**(1/4)) + log(sqrt(x) + (-a/b)**(1/4))/(2*a*(-a/b)**(1/4)) - atan(
sqrt(x)/(-a/b)**(1/4))/(a*(-a/b)**(1/4)) - 2/(a*sqrt(x)), True)) + 3*c**2*d*Piecewise((zoo/sqrt(x), Eq(a, 0) &
Eq(b, 0)), (2*x**(3/2)/(3*a), Eq(b, 0)), (-2/(b*sqrt(x)), Eq(a, 0)), (log(sqrt(x) - (-a/b)**(1/4))/(2*b*(-a/b
)**(1/4)) - log(sqrt(x) + (-a/b)**(1/4))/(2*b*(-a/b)**(1/4)) + atan(sqrt(x)/(-a/b)**(1/4))/(b*(-a/b)**(1/4)),
True)) + 3*c*d**2*Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(7/2)/(7*a), Eq(b, 0)), (2*x**(3/2)/(3*
b), Eq(a, 0)), (-a*log(sqrt(x) - (-a/b)**(1/4))/(2*b**2*(-a/b)**(1/4)) + a*log(sqrt(x) + (-a/b)**(1/4))/(2*b**
2*(-a/b)**(1/4)) - a*atan(sqrt(x)/(-a/b)**(1/4))/(b**2*(-a/b)**(1/4)) + 2*x**(3/2)/(3*b), True)) + d**3*Piecew
ise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(11/2)/(11*a), Eq(b, 0)), (2*x**(7/2)/(7*b), Eq(a, 0)), (a**2*l
og(sqrt(x) - (-a/b)**(1/4))/(2*b**3*(-a/b)**(1/4)) - a**2*log(sqrt(x) + (-a/b)**(1/4))/(2*b**3*(-a/b)**(1/4))
+ a**2*atan(sqrt(x)/(-a/b)**(1/4))/(b**3*(-a/b)**(1/4)) - 2*a*x**(3/2)/(3*b**2) + 2*x**(7/2)/(7*b), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{a \sqrt {x}} + \frac {2 \, {\left (3 \, b d^{3} x^{\frac {7}{2}} + 7 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x^{\frac {3}{2}}\right )}}{21 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, a b^{2}} \]

[In]

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

[Out]

-2*c^3/(a*sqrt(x)) + 2/21*(3*b*d^3*x^(7/2) + 7*(3*b*c*d^2 - a*d^3)*x^(3/2))/b^2 - 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(s
qrt(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)))/(a*b^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (207) = 414\).

Time = 0.31 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{a \sqrt {x}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{5}} + \frac {2 \, {\left (3 \, b^{6} d^{3} x^{\frac {7}{2}} + 21 \, b^{6} c d^{2} x^{\frac {3}{2}} - 7 \, a b^{5} d^{3} x^{\frac {3}{2}}\right )}}{21 \, b^{7}} \]

[In]

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

[Out]

-2*c^3/(a*sqrt(x)) - 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^2*b^5) - 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^2*b^5) + 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^2*b^5) - 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^2*b^5) + 2/21*(3*b^6*d^3*x^(7/2) + 21*b^6*c*d^2*x^(3/2) - 7*a*b^5*d^3*x^(3/2))/b^7

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.04 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx=\frac {2\,d^3\,x^{7/2}}{7\,b}-\frac {2\,c^3}{a\,\sqrt {x}}-x^{3/2}\,\left (\frac {2\,a\,d^3}{3\,b^2}-\frac {2\,c\,d^2}{b}\right )-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{10}\,b^8\,d^6-96\,a^9\,b^9\,c\,d^5+240\,a^8\,b^{10}\,c^2\,d^4-320\,a^7\,b^{11}\,c^3\,d^3+240\,a^6\,b^{12}\,c^4\,d^2-96\,a^5\,b^{13}\,c^5\,d+16\,a^4\,b^{14}\,c^6\right )}{{\left (-a\right )}^{5/4}\,b^{11/4}\,\left (-16\,a^{12}\,b^5\,d^9+144\,a^{11}\,b^6\,c\,d^8-576\,a^{10}\,b^7\,c^2\,d^7+1344\,a^9\,b^8\,c^3\,d^6-2016\,a^8\,b^9\,c^4\,d^5+2016\,a^7\,b^{10}\,c^5\,d^4-1344\,a^6\,b^{11}\,c^6\,d^3+576\,a^5\,b^{12}\,c^7\,d^2-144\,a^4\,b^{13}\,c^8\,d+16\,a^3\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{10}\,b^8\,d^6-96\,a^9\,b^9\,c\,d^5+240\,a^8\,b^{10}\,c^2\,d^4-320\,a^7\,b^{11}\,c^3\,d^3+240\,a^6\,b^{12}\,c^4\,d^2-96\,a^5\,b^{13}\,c^5\,d+16\,a^4\,b^{14}\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{5/4}\,b^{11/4}\,\left (-16\,a^{12}\,b^5\,d^9+144\,a^{11}\,b^6\,c\,d^8-576\,a^{10}\,b^7\,c^2\,d^7+1344\,a^9\,b^8\,c^3\,d^6-2016\,a^8\,b^9\,c^4\,d^5+2016\,a^7\,b^{10}\,c^5\,d^4-1344\,a^6\,b^{11}\,c^6\,d^3+576\,a^5\,b^{12}\,c^7\,d^2-144\,a^4\,b^{13}\,c^8\,d+16\,a^3\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{5/4}\,b^{11/4}} \]

[In]

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

[Out]

(2*d^3*x^(7/2))/(7*b) - (2*c^3)/(a*x^(1/2)) - x^(3/2)*((2*a*d^3)/(3*b^2) - (2*c*d^2)/b) - (atan((x^(1/2)*(a*d
- b*c)^3*(16*a^4*b^14*c^6 + 16*a^10*b^8*d^6 - 96*a^5*b^13*c^5*d - 96*a^9*b^9*c*d^5 + 240*a^6*b^12*c^4*d^2 - 32
0*a^7*b^11*c^3*d^3 + 240*a^8*b^10*c^2*d^4))/((-a)^(5/4)*b^(11/4)*(16*a^3*b^14*c^9 - 16*a^12*b^5*d^9 - 144*a^4*
b^13*c^8*d + 144*a^11*b^6*c*d^8 + 576*a^5*b^12*c^7*d^2 - 1344*a^6*b^11*c^6*d^3 + 2016*a^7*b^10*c^5*d^4 - 2016*
a^8*b^9*c^4*d^5 + 1344*a^9*b^8*c^3*d^6 - 576*a^10*b^7*c^2*d^7)))*(a*d - b*c)^3)/((-a)^(5/4)*b^(11/4)) - (atan(
(x^(1/2)*(a*d - b*c)^3*(16*a^4*b^14*c^6 + 16*a^10*b^8*d^6 - 96*a^5*b^13*c^5*d - 96*a^9*b^9*c*d^5 + 240*a^6*b^1
2*c^4*d^2 - 320*a^7*b^11*c^3*d^3 + 240*a^8*b^10*c^2*d^4)*1i)/((-a)^(5/4)*b^(11/4)*(16*a^3*b^14*c^9 - 16*a^12*b
^5*d^9 - 144*a^4*b^13*c^8*d + 144*a^11*b^6*c*d^8 + 576*a^5*b^12*c^7*d^2 - 1344*a^6*b^11*c^6*d^3 + 2016*a^7*b^1
0*c^5*d^4 - 2016*a^8*b^9*c^4*d^5 + 1344*a^9*b^8*c^3*d^6 - 576*a^10*b^7*c^2*d^7)))*(a*d - b*c)^3*1i)/((-a)^(5/4
)*b^(11/4))