\(\int \frac {(a+b x^2)^{3/2}}{x^4 (c+d x)^3} \, dx\) [1102]

Optimal result

Integrand size = 22, antiderivative size = 286 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=-\frac {a \sqrt {a+b x^2}}{3 c^3 x^3}+\frac {3 a d \sqrt {a+b x^2}}{2 c^4 x^2}-\frac {2 \left (2 b c^2+9 a d^2\right ) \sqrt {a+b x^2}}{3 c^5 x}-\frac {d \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{2 c^4 (c+d x)^2}-\frac {d \left (3 b c^2+8 a d^2\right ) \sqrt {a+b x^2}}{2 c^5 (c+d x)}-\frac {\left (2 b^2 c^4+19 a b c^2 d^2+20 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 c^6 \sqrt {b c^2+a d^2}}+\frac {\sqrt {a} d \left (9 b c^2+20 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^6} \] Output:

-1/3*a*(b*x^2+a)^(1/2)/c^3/x^3+3/2*a*d*(b*x^2+a)^(1/2)/c^4/x^2-2/3*(9*a*d^ 
2+2*b*c^2)*(b*x^2+a)^(1/2)/c^5/x-1/2*d*(a*d^2+b*c^2)*(b*x^2+a)^(1/2)/c^4/( 
d*x+c)^2-1/2*d*(8*a*d^2+3*b*c^2)*(b*x^2+a)^(1/2)/c^5/(d*x+c)-1/2*(20*a^2*d 
^4+19*a*b*c^2*d^2+2*b^2*c^4)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x 
^2+a)^(1/2))/c^6/(a*d^2+b*c^2)^(1/2)+1/2*a^(1/2)*d*(20*a*d^2+9*b*c^2)*arct 
anh((b*x^2+a)^(1/2)/a^(1/2))/c^6
 

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (b c^2 x^2 \left (8 c^2+28 c d x+17 d^2 x^2\right )+a \left (2 c^4-5 c^3 d x+20 c^2 d^2 x^2+90 c d^3 x^3+60 d^4 x^4\right )\right )}{x^3 (c+d x)^2}+\frac {6 \left (2 b^2 c^4+19 a b c^2 d^2+20 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+6 \sqrt {a} d \left (9 b c^2+20 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{6 c^6} \] Input:

Integrate[(a + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x]
 

Output:

-1/6*((c*Sqrt[a + b*x^2]*(b*c^2*x^2*(8*c^2 + 28*c*d*x + 17*d^2*x^2) + a*(2 
*c^4 - 5*c^3*d*x + 20*c^2*d^2*x^2 + 90*c*d^3*x^3 + 60*d^4*x^4)))/(x^3*(c + 
 d*x)^2) + (6*(2*b^2*c^4 + 19*a*b*c^2*d^2 + 20*a^2*d^4)*ArcTan[(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]*d*(9*b*c^2 + 20*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2] 
)/Sqrt[a]])/c^6
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(693\) vs. \(2(286)=572\).

Time = 1.76 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.42, 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.

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

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {10 d^4 \left (a+b x^2\right )^{3/2}}{c^6 (c+d x)}-\frac {10 d^3 \left (a+b x^2\right )^{3/2}}{c^6 x}+\frac {4 d^4 \left (a+b x^2\right )^{3/2}}{c^5 (c+d x)^2}+\frac {6 d^2 \left (a+b x^2\right )^{3/2}}{c^5 x^2}+\frac {d^4 \left (a+b x^2\right )^{3/2}}{c^4 (c+d x)^3}-\frac {3 d \left (a+b x^2\right )^{3/2}}{c^4 x^3}+\frac {\left (a+b x^2\right )^{3/2}}{c^3 x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10 a^{3/2} d^3 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^6}-\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{c^3}+\frac {9 a \sqrt {b} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{c^5}+\frac {9 \sqrt {a} b d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^4}-\frac {10 \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^6}+\frac {6 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+2 b c^2\right )}{c^5}-\frac {5 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right )}{c^5}-\frac {3 b \left (a d^2+2 b c^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 c^4 \sqrt {a d^2+b c^2}}+\frac {12 b \sqrt {a d^2+b c^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^4}-\frac {10 a d^3 \sqrt {a+b x^2}}{c^6}-\frac {4 d^3 \left (a+b x^2\right )^{3/2}}{c^5 (c+d x)}-\frac {6 d^2 \left (a+b x^2\right )^{3/2}}{c^5 x}+\frac {9 b d^2 x \sqrt {a+b x^2}}{c^5}-\frac {6 b d \sqrt {a+b x^2} (2 c-d x)}{c^5}-\frac {d^3 \left (a+b x^2\right )^{3/2}}{2 c^4 (c+d x)^2}+\frac {3 d \left (a+b x^2\right )^{3/2}}{2 c^4 x^2}+\frac {3 b d \sqrt {a+b x^2} (2 c+d x)}{2 c^4 (c+d x)}-\frac {9 b d \sqrt {a+b x^2}}{2 c^4}-\frac {b \sqrt {a+b x^2}}{c^3 x}-\frac {\left (a+b x^2\right )^{3/2}}{3 c^3 x^3}+\frac {5 d \sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{c^6}\)

Input:

Int[(a + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x]
 

Output:

(-9*b*d*Sqrt[a + b*x^2])/(2*c^4) - (10*a*d^3*Sqrt[a + b*x^2])/c^6 - (b*Sqr 
t[a + b*x^2])/(c^3*x) + (9*b*d^2*x*Sqrt[a + b*x^2])/c^5 - (6*b*d*(2*c - d* 
x)*Sqrt[a + b*x^2])/c^5 + (3*b*d*(2*c + d*x)*Sqrt[a + b*x^2])/(2*c^4*(c + 
d*x)) + (5*d*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b*x^2])/c^6 - (a + b*x 
^2)^(3/2)/(3*c^3*x^3) + (3*d*(a + b*x^2)^(3/2))/(2*c^4*x^2) - (6*d^2*(a + 
b*x^2)^(3/2))/(c^5*x) - (d^3*(a + b*x^2)^(3/2))/(2*c^4*(c + d*x)^2) - (4*d 
^3*(a + b*x^2)^(3/2))/(c^5*(c + d*x)) - (2*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/c^3 + (9*a*Sqrt[b]*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] 
)/c^5 + (6*Sqrt[b]*(2*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]) 
/c^5 - (5*Sqrt[b]*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] 
)/c^5 + (12*b*Sqrt[b*c^2 + a*d^2]*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^ 
2]*Sqrt[a + b*x^2])])/c^4 - (10*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d - b*c*x 
)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/c^6 - (3*b*(2*b*c^2 + a*d^2)*Arc 
Tanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(2*c^4*Sqrt[b*c 
^2 + a*d^2]) + (9*Sqrt[a]*b*d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*c^4) + 
(10*a^(3/2)*d^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^6
 

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. \(934\) vs. \(2(250)=500\).

Time = 0.53 (sec) , antiderivative size = 935, normalized size of antiderivative = 3.27

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (36 a \,d^{2} x^{2}+8 b \,c^{2} x^{2}-9 a d x c +2 a \,c^{2}\right )}{6 c^{5} x^{3}}+\frac {\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^{3}}+8 a \left (a \,d^{2}+b \,c^{2}\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 )+\frac {\sqrt {a}\, d \left (20 a \,d^{2}+9 b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}-\frac {4 d a \left (5 a \,d^{2}+3 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^{5}}\)	\(935\)
default	\(\text {Expression too large to display}\)	\(3175\)

Input:

int((b*x^2+a)^(3/2)/x^4/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/6*(b*x^2+a)^(1/2)*(36*a*d^2*x^2+8*b*c^2*x^2-9*a*c*d*x+2*a*c^2)/c^5/x^3+ 
1/2/c^5*(2*c*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^3*(-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)))+8*a*(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)))+ 
a^(1/2)/c*d*(20*a*d^2+9*b*c^2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-4/c*d 
*a*(5*a*d^2+3*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. 542 vs. \(2 (251) = 502\).

Time = 0.49 (sec) , antiderivative size = 2238, normalized size of antiderivative = 7.83 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[1/12*(3*((2*b^2*c^4*d^2 + 19*a*b*c^2*d^4 + 20*a^2*d^6)*x^5 + 2*(2*b^2*c^5 
*d + 19*a*b*c^3*d^3 + 20*a^2*c*d^5)*x^4 + (2*b^2*c^6 + 19*a*b*c^4*d^2 + 20 
*a^2*c^2*d^4)*x^3)*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*((9*b^2*c^4*d^3 + 29*a*b*c^2*d 
^5 + 20*a^2*d^7)*x^5 + 2*(9*b^2*c^5*d^2 + 29*a*b*c^3*d^4 + 20*a^2*c*d^6)*x 
^4 + (9*b^2*c^6*d + 29*a*b*c^4*d^3 + 20*a^2*c^2*d^5)*x^3)*sqrt(a)*log(-(b* 
x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*a*b*c^7 + 2*a^2*c^5*d^2 
 + (17*b^2*c^5*d^2 + 77*a*b*c^3*d^4 + 60*a^2*c*d^6)*x^4 + 2*(14*b^2*c^6*d 
+ 59*a*b*c^4*d^3 + 45*a^2*c^2*d^5)*x^3 + 4*(2*b^2*c^7 + 7*a*b*c^5*d^2 + 5* 
a^2*c^3*d^4)*x^2 - 5*(a*b*c^6*d + a^2*c^4*d^3)*x)*sqrt(b*x^2 + a))/((b*c^8 
*d^2 + a*c^6*d^4)*x^5 + 2*(b*c^9*d + a*c^7*d^3)*x^4 + (b*c^10 + a*c^8*d^2) 
*x^3), -1/12*(6*((2*b^2*c^4*d^2 + 19*a*b*c^2*d^4 + 20*a^2*d^6)*x^5 + 2*(2* 
b^2*c^5*d + 19*a*b*c^3*d^3 + 20*a^2*c*d^5)*x^4 + (2*b^2*c^6 + 19*a*b*c^4*d 
^2 + 20*a^2*c^2*d^4)*x^3)*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*((9*b^2*c^4*d^3 + 29*a*b*c^2*d^5 + 20*a^2*d^7)*x^5 + 2*(9*b^2*c^5* 
d^2 + 29*a*b*c^3*d^4 + 20*a^2*c*d^6)*x^4 + (9*b^2*c^6*d + 29*a*b*c^4*d^3 + 
 20*a^2*c^2*d^5)*x^3)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2* 
a)/x^2) + 2*(2*a*b*c^7 + 2*a^2*c^5*d^2 + (17*b^2*c^5*d^2 + 77*a*b*c^3*d...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{4} \left (c + d x\right )^{3}}\, dx \] Input:

integrate((b*x**2+a)**(3/2)/x**4/(d*x+c)**3,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*x**2)**(3/2)/(x**4*(c + d*x)**3), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{3} x^{4}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x + c)^3*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(3/2)/x^4/(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:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^4\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int((a + b*x^2)^(3/2)/(x^4*(c + d*x)^3),x)
 

Output:

int((a + b*x^2)^(3/2)/(x^4*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1596, normalized size of antiderivative = 5.58 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(3/2)/x^4/(d*x+c)^3,x)
 

Output:

(120*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**4*x**3 + 240*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**5*x**4 + 120*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**6*x**5 + 114*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**2*x**3 + 228*sqrt(a*d**2 + b*c**2)*lo 
g(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d**3*x**4 
 + 114*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**4*x**5 + 12*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)*b**2*c**6*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)*b**2* 
c**5*d*x**4 + 12*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)*b**2*c**4*d**2*x**5 - 120*sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*a**2*c**2*d**4*x**3 - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a** 
2*c*d**5*x**4 - 120*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**6*x**5 - 11 
4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**4*d**2*x**3 - 228*sqrt(a*d**2 
+ b*c**2)*log(c + d*x)*a*b*c**3*d**3*x**4 - 114*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a*b*c**2*d**4*x**5 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c 
**6*x**3 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**5*d*x**4 - 12*sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4*d**2*x**5 - 4*sqrt(a + b*x**2...