\(\int \frac {(a+b x^2)^{3/2}}{x^3 (c+d x)^3} \, dx\) [1101]

Optimal result

Integrand size = 22, antiderivative size = 233 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=-\frac {a \sqrt {a+b x^2}}{2 c^3 x^2}+\frac {3 a d \sqrt {a+b x^2}}{c^4 x}+\frac {\left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{2 c^3 (c+d x)^2}+\frac {\left (b c^2+6 a d^2\right ) \sqrt {a+b x^2}}{2 c^4 (c+d x)}+\frac {3 a d \left (3 b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^5 \sqrt {b c^2+a d^2}}-\frac {3 \sqrt {a} \left (b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^5} \] Output:

-1/2*a*(b*x^2+a)^(1/2)/c^3/x^2+3*a*d*(b*x^2+a)^(1/2)/c^4/x+1/2*(a*d^2+b*c^ 
2)*(b*x^2+a)^(1/2)/c^3/(d*x+c)^2+1/2*(6*a*d^2+b*c^2)*(b*x^2+a)^(1/2)/c^4/( 
d*x+c)+3/2*a*d*(4*a*d^2+3*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/ 
(b*x^2+a)^(1/2))/c^5/(a*d^2+b*c^2)^(1/2)-3/2*a^(1/2)*(4*a*d^2+b*c^2)*arcta 
nh((b*x^2+a)^(1/2)/a^(1/2))/c^5
 

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\frac {\frac {c \sqrt {a+b x^2} \left (b c^2 x^2 (2 c+d x)+a \left (-c^3+4 c^2 d x+18 c d^2 x^2+12 d^3 x^3\right )\right )}{x^2 (c+d x)^2}+\frac {6 a d \left (3 b c^2+4 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+6 \sqrt {a} \left (b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^5} \] Input:

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

Output:

((c*Sqrt[a + b*x^2]*(b*c^2*x^2*(2*c + d*x) + a*(-c^3 + 4*c^2*d*x + 18*c*d^ 
2*x^2 + 12*d^3*x^3)))/(x^2*(c + d*x)^2) + (6*a*d*(3*b*c^2 + 4*a*d^2)*ArcTa 
n[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/Sqrt[-( 
b*c^2) - a*d^2] + 6*Sqrt[a]*(b*c^2 + 4*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a 
+ b*x^2])/Sqrt[a]])/(2*c^5)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(666\) vs. \(2(233)=466\).

Time = 1.61 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (-\frac {6 d^3 \left (a+b x^2\right )^{3/2}}{c^5 (c+d x)}+\frac {6 d^2 \left (a+b x^2\right )^{3/2}}{c^5 x}-\frac {3 d^3 \left (a+b x^2\right )^{3/2}}{c^4 (c+d x)^2}-\frac {3 d \left (a+b x^2\right )^{3/2}}{c^4 x^2}-\frac {d^3 \left (a+b x^2\right )^{3/2}}{c^3 (c+d x)^3}+\frac {\left (a+b x^2\right )^{3/2}}{c^3 x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 a^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^5}+\frac {3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{c^2 d}-\frac {9 a \sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 c^4}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^3}+\frac {6 \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5 d}-\frac {9 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right )}{2 c^4 d}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right )}{c^4 d}+\frac {3 b \left (a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 c^3 d \sqrt {a d^2+b c^2}}-\frac {9 b \sqrt {a d^2+b c^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 d}+\frac {6 a d^2 \sqrt {a+b x^2}}{c^5}+\frac {3 d^2 \left (a+b x^2\right )^{3/2}}{c^4 (c+d x)}+\frac {3 d \left (a+b x^2\right )^{3/2}}{c^4 x}-\frac {9 b d x \sqrt {a+b x^2}}{2 c^4}+\frac {9 b \sqrt {a+b x^2} (2 c-d x)}{2 c^4}+\frac {d^2 \left (a+b x^2\right )^{3/2}}{2 c^3 (c+d x)^2}-\frac {3 b \sqrt {a+b x^2} (2 c+d x)}{2 c^3 (c+d x)}-\frac {\left (a+b x^2\right )^{3/2}}{2 c^3 x^2}+\frac {3 b \sqrt {a+b x^2}}{2 c^3}-\frac {3 \sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{c^5}\)

Input:

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

Output:

(3*b*Sqrt[a + b*x^2])/(2*c^3) + (6*a*d^2*Sqrt[a + b*x^2])/c^5 - (9*b*d*x*S 
qrt[a + b*x^2])/(2*c^4) + (9*b*(2*c - d*x)*Sqrt[a + b*x^2])/(2*c^4) - (3*b 
*(2*c + d*x)*Sqrt[a + b*x^2])/(2*c^3*(c + d*x)) - (3*(2*(b*c^2 + a*d^2) - 
b*c*d*x)*Sqrt[a + b*x^2])/c^5 - (a + b*x^2)^(3/2)/(2*c^3*x^2) + (3*d*(a + 
b*x^2)^(3/2))/(c^4*x) + (d^2*(a + b*x^2)^(3/2))/(2*c^3*(c + d*x)^2) + (3*d 
^2*(a + b*x^2)^(3/2))/(c^4*(c + d*x)) + (3*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/(c^2*d) - (9*a*Sqrt[b]*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2 
]])/(2*c^4) - (9*Sqrt[b]*(2*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b* 
x^2]])/(2*c^4*d) + (3*Sqrt[b]*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/(c^4*d) - (9*b*Sqrt[b*c^2 + a*d^2]*ArcTanh[(a*d - b*c*x)/(Sq 
rt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^3*d) + (6*(b*c^2 + a*d^2)^(3/2)*Ar 
cTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^5*d) + (3*b 
*(2*b*c^2 + a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x 
^2])])/(2*c^3*d*Sqrt[b*c^2 + a*d^2]) - (3*Sqrt[a]*b*ArcTanh[Sqrt[a + b*x^2 
]/Sqrt[a]])/(2*c^3) - (6*a^(3/2)*d^2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^5
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(925\) vs. \(2(203)=406\).

Time = 0.53 (sec) , antiderivative size = 926, normalized size of antiderivative = 3.97

method	result	size
risch	\(-\frac {a \sqrt {b \,x^{2}+a}\, \left (-6 d x +c \right )}{2 c^{4} x^{2}}-\frac {\frac {2 \left (3 a^{2} d^{4}+2 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}+\frac {3 \sqrt {a}\, \left (4 a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}+\frac {2 c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}-\frac {4 a \left (3 a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 c^{4}}\)	\(926\)
default	\(\text {Expression too large to display}\)	\(3066\)

Input:

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

Output:

-1/2*a*(b*x^2+a)^(1/2)*(-6*d*x+c)/c^4/x^2-1/2/c^4*(2*(3*a^2*d^4+2*a*b*c^2* 
d^2-b^2*c^4)/d^3*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d 
)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*l 
n((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c 
/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+3*a^(1/2)/c*(4*a 
*d^2+b*c^2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+2*c*(a^2*d^4+2*a*b*c^2*d 
^2+b^2*c^4)/d^4*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*(x+ 
c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+3/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^ 
2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d 
^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d 
)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d 
^2)^(1/2))/(x+c/d)))+1/2*b/(a*d^2+b*c^2)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln( 
(2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d 
)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))-4/c*a*(3*a*d^2+b*c 
^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*(( 
a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1 
/2))/(x+c/d)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (204) = 408\).

Time = 0.33 (sec) , antiderivative size = 1931, normalized size of antiderivative = 8.29 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(3*((3*a*b*c^2*d^3 + 4*a^2*d^5)*x^4 + 2*(3*a*b*c^3*d^2 + 4*a^2*c*d^4) 
*x^3 + (3*a*b*c^4*d + 4*a^2*c^2*d^3)*x^2)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c 
*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a* 
d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 3*((b^2*c 
^4*d^2 + 5*a*b*c^2*d^4 + 4*a^2*d^6)*x^4 + 2*(b^2*c^5*d + 5*a*b*c^3*d^3 + 4 
*a^2*c*d^5)*x^3 + (b^2*c^6 + 5*a*b*c^4*d^2 + 4*a^2*c^2*d^4)*x^2)*sqrt(a)*l 
og(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*b*c^6 + a^2*c^4* 
d^2 - (b^2*c^5*d + 13*a*b*c^3*d^3 + 12*a^2*c*d^5)*x^3 - 2*(b^2*c^6 + 10*a* 
b*c^4*d^2 + 9*a^2*c^2*d^4)*x^2 - 4*(a*b*c^5*d + a^2*c^3*d^3)*x)*sqrt(b*x^2 
 + a))/((b*c^7*d^2 + a*c^5*d^4)*x^4 + 2*(b*c^8*d + a*c^6*d^3)*x^3 + (b*c^9 
 + a*c^7*d^2)*x^2), 1/4*(6*((3*a*b*c^2*d^3 + 4*a^2*d^5)*x^4 + 2*(3*a*b*c^3 
*d^2 + 4*a^2*c*d^4)*x^3 + (3*a*b*c^4*d + 4*a^2*c^2*d^3)*x^2)*sqrt(-b*c^2 - 
 a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 
 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + 3*((b^2*c^4*d^2 + 5*a*b*c^2*d^4 + 
 4*a^2*d^6)*x^4 + 2*(b^2*c^5*d + 5*a*b*c^3*d^3 + 4*a^2*c*d^5)*x^3 + (b^2*c 
^6 + 5*a*b*c^4*d^2 + 4*a^2*c^2*d^4)*x^2)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^ 
2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*b*c^6 + a^2*c^4*d^2 - (b^2*c^5*d + 13*a* 
b*c^3*d^3 + 12*a^2*c*d^5)*x^3 - 2*(b^2*c^6 + 10*a*b*c^4*d^2 + 9*a^2*c^2*d^ 
4)*x^2 - 4*(a*b*c^5*d + a^2*c^3*d^3)*x)*sqrt(b*x^2 + a))/((b*c^7*d^2 + a*c 
^5*d^4)*x^4 + 2*(b*c^8*d + a*c^6*d^3)*x^3 + (b*c^9 + a*c^7*d^2)*x^2), 1...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (c + d x\right )^{3}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{3} x^{3}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^3\,{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1270, normalized size of antiderivative = 5.45 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(24*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**2*c**2*d**3*x**2 + 48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a 
+ b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*c*d**4*x**3 + 24*sqrt( 
a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c 
*x)*a**2*d**5*x**4 + 18*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt 
(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**4*d*x**2 + 36*sqrt(a*d**2 + b*c**2 
)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d* 
*2*x**3 + 18*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b 
*c**2) - a*d + b*c*x)*a*b*c**2*d**3*x**4 - 24*sqrt(a*d**2 + b*c**2)*log(c 
+ d*x)*a**2*c**2*d**3*x**2 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*c* 
d**4*x**3 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**5*x**4 - 18*sqrt 
(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**4*d*x**2 - 36*sqrt(a*d**2 + b*c**2)* 
log(c + d*x)*a*b*c**3*d**2*x**3 - 18*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a* 
b*c**2*d**3*x**4 - 2*sqrt(a + b*x**2)*a**2*c**4*d**2 + 8*sqrt(a + b*x**2)* 
a**2*c**3*d**3*x + 36*sqrt(a + b*x**2)*a**2*c**2*d**4*x**2 + 24*sqrt(a + b 
*x**2)*a**2*c*d**5*x**3 - 2*sqrt(a + b*x**2)*a*b*c**6 + 8*sqrt(a + b*x**2) 
*a*b*c**5*d*x + 40*sqrt(a + b*x**2)*a*b*c**4*d**2*x**2 + 26*sqrt(a + b*x** 
2)*a*b*c**3*d**3*x**3 + 4*sqrt(a + b*x**2)*b**2*c**6*x**2 + 2*sqrt(a + b*x 
**2)*b**2*c**5*d*x**3 + 12*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*c* 
*2*d**4*x**2 + 24*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*c*d**5*x...