\(\int \frac {(a+b x^2)^{5/2}}{x^6 (c+d x)} \, dx\) [1136]

Optimal result

Integrand size = 22, antiderivative size = 284 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=-\frac {a^2 \sqrt {a+b x^2}}{5 c x^5}+\frac {a^2 d \sqrt {a+b x^2}}{4 c^2 x^4}-\frac {a \left (11 b c^2+5 a d^2\right ) \sqrt {a+b x^2}}{15 c^3 x^3}+\frac {a d \left (9 b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{8 c^4 x^2}-\frac {\left (23 b^2 c^4+35 a b c^2 d^2+15 a^2 d^4\right ) \sqrt {a+b x^2}}{15 c^5 x}-\frac {\left (b c^2+a d^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^6}+\frac {\sqrt {a} d \left (15 b^2 c^4+20 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 c^6} \] Output:

-1/5*a^2*(b*x^2+a)^(1/2)/c/x^5+1/4*a^2*d*(b*x^2+a)^(1/2)/c^2/x^4-1/15*a*(5 
*a*d^2+11*b*c^2)*(b*x^2+a)^(1/2)/c^3/x^3+1/8*a*d*(4*a*d^2+9*b*c^2)*(b*x^2+ 
a)^(1/2)/c^4/x^2-1/15*(15*a^2*d^4+35*a*b*c^2*d^2+23*b^2*c^4)*(b*x^2+a)^(1/ 
2)/c^5/x-(a*d^2+b*c^2)^(5/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x 
^2+a)^(1/2))/c^6+1/8*a^(1/2)*d*(8*a^2*d^4+20*a*b*c^2*d^2+15*b^2*c^4)*arcta 
nh((b*x^2+a)^(1/2)/a^(1/2))/c^6
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\frac {-\frac {c \sqrt {a+b x^2} \left (184 b^2 c^4 x^4+a b c^2 x^2 \left (88 c^2-135 c d x+280 d^2 x^2\right )+2 a^2 \left (12 c^4-15 c^3 d x+20 c^2 d^2 x^2-30 c d^3 x^3+60 d^4 x^4\right )\right )}{x^5}+240 \left (-b c^2-a d^2\right )^{5/2} \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )+15 \sqrt {a} d \left (15 b^2 c^4+20 a b c^2 d^2+8 a^2 d^4\right ) \log (x)-15 \sqrt {a} d \left (15 b^2 c^4+20 a b c^2 d^2+8 a^2 d^4\right ) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{120 c^6} \] Input:

Integrate[(a + b*x^2)^(5/2)/(x^6*(c + d*x)),x]
 

Output:

(-((c*Sqrt[a + b*x^2]*(184*b^2*c^4*x^4 + a*b*c^2*x^2*(88*c^2 - 135*c*d*x + 
 280*d^2*x^2) + 2*a^2*(12*c^4 - 15*c^3*d*x + 20*c^2*d^2*x^2 - 30*c*d^3*x^3 
 + 60*d^4*x^4)))/x^5) + 240*(-(b*c^2) - a*d^2)^(5/2)*ArcTan[(Sqrt[-(b*c^2) 
 - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2])] + 15*Sqrt[a]*d*(15*b 
^2*c^4 + 20*a*b*c^2*d^2 + 8*a^2*d^4)*Log[x] - 15*Sqrt[a]*d*(15*b^2*c^4 + 2 
0*a*b*c^2*d^2 + 8*a^2*d^4)*Log[-Sqrt[a] + Sqrt[a + b*x^2]])/(120*c^6)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(797\) vs. \(2(284)=568\).

Time = 1.78 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.81, 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.

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

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (b x^2+a\right )^{3/2} d^5}{3 c^6}+\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^5}{c^6}-\frac {a^2 \sqrt {b x^2+a} d^5}{c^6}-\frac {\left (b x^2+a\right )^{5/2} d^4}{c^5 x}+\frac {5 b x \left (b x^2+a\right )^{3/2} d^4}{4 c^5}+\frac {15 a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^4}{8 c^5}+\frac {15 a b x \sqrt {b x^2+a} d^4}{8 c^5}+\frac {\left (b x^2+a\right )^{5/2} d^3}{2 c^4 x^2}+\frac {\left (4 \left (b c^2+a d^2\right )-3 b c d x\right ) \left (b x^2+a\right )^{3/2} d^3}{12 c^6}-\frac {5 b \left (b x^2+a\right )^{3/2} d^3}{6 c^4}+\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^3}{2 c^4}-\frac {5 a b \sqrt {b x^2+a} d^3}{2 c^4}-\frac {\left (b x^2+a\right )^{5/2} d^2}{3 c^3 x^3}-\frac {5 b \left (b x^2+a\right )^{3/2} d^2}{3 c^3 x}+\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^2}{2 c^3}+\frac {5 b^2 x \sqrt {b x^2+a} d^2}{2 c^3}+\frac {\left (b x^2+a\right )^{5/2} d}{4 c^2 x^4}+\frac {5 b \left (b x^2+a\right )^{3/2} d}{8 c^2 x^2}+\frac {15 \sqrt {a} b^2 \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d}{8 c^2}+\frac {\left (8 \left (b c^2+a d^2\right )^2-b c d \left (4 b c^2+7 a d^2\right ) x\right ) \sqrt {b x^2+a} d}{8 c^6}-\frac {15 b^2 \sqrt {b x^2+a} d}{8 c^2}-\frac {\left (b x^2+a\right )^{5/2}}{5 c x^5}-\frac {b \left (b x^2+a\right )^{3/2}}{3 c x^3}-\frac {\sqrt {b} \left (8 b^2 c^4+20 a b d^2 c^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{8 c^5}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{c}-\frac {\left (b c^2+a d^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{c^6}-\frac {b^2 \sqrt {b x^2+a}}{c x}\)

Input:

Int[(a + b*x^2)^(5/2)/(x^6*(c + d*x)),x]
 

Output:

(-15*b^2*d*Sqrt[a + b*x^2])/(8*c^2) - (5*a*b*d^3*Sqrt[a + b*x^2])/(2*c^4) 
- (a^2*d^5*Sqrt[a + b*x^2])/c^6 - (b^2*Sqrt[a + b*x^2])/(c*x) + (5*b^2*d^2 
*x*Sqrt[a + b*x^2])/(2*c^3) + (15*a*b*d^4*x*Sqrt[a + b*x^2])/(8*c^5) + (d* 
(8*(b*c^2 + a*d^2)^2 - b*c*d*(4*b*c^2 + 7*a*d^2)*x)*Sqrt[a + b*x^2])/(8*c^ 
6) - (5*b*d^3*(a + b*x^2)^(3/2))/(6*c^4) - (a*d^5*(a + b*x^2)^(3/2))/(3*c^ 
6) - (b*(a + b*x^2)^(3/2))/(3*c*x^3) + (5*b*d*(a + b*x^2)^(3/2))/(8*c^2*x^ 
2) - (5*b*d^2*(a + b*x^2)^(3/2))/(3*c^3*x) + (5*b*d^4*x*(a + b*x^2)^(3/2)) 
/(4*c^5) + (d^3*(4*(b*c^2 + a*d^2) - 3*b*c*d*x)*(a + b*x^2)^(3/2))/(12*c^6 
) - (a + b*x^2)^(5/2)/(5*c*x^5) + (d*(a + b*x^2)^(5/2))/(4*c^2*x^4) - (d^2 
*(a + b*x^2)^(5/2))/(3*c^3*x^3) + (d^3*(a + b*x^2)^(5/2))/(2*c^4*x^2) - (d 
^4*(a + b*x^2)^(5/2))/(c^5*x) + (b^(5/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^ 
2]])/c + (5*a*b^(3/2)*d^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^3) + 
(15*a^2*Sqrt[b]*d^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*c^5) - (Sqrt[ 
b]*(8*b^2*c^4 + 20*a*b*c^2*d^2 + 15*a^2*d^4)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + 
b*x^2]])/(8*c^5) - ((b*c^2 + a*d^2)^(5/2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^ 
2 + a*d^2]*Sqrt[a + b*x^2])])/c^6 + (15*Sqrt[a]*b^2*d*ArcTanh[Sqrt[a + b*x 
^2]/Sqrt[a]])/(8*c^2) + (5*a^(3/2)*b*d^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]) 
/(2*c^4) + (a^(5/2)*d^5*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 [A] (verified)

Time = 0.51 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.24

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (120 a^{2} d^{4} x^{4}+280 x^{4} a b \,c^{2} d^{2}+184 x^{4} b^{2} c^{4}-60 a^{2} c \,d^{3} x^{3}-135 a b \,c^{3} d \,x^{3}+40 a^{2} c^{2} d^{2} x^{2}+88 a b \,c^{4} x^{2}-30 a^{2} d \,c^{3} x +24 a^{2} c^{4}\right )}{120 c^{5} x^{5}}+\frac {-\frac {8 \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c d \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {d \sqrt {a}\, \left (8 a^{2} d^{4}+20 b \,c^{2} d^{2} a +15 b^{2} c^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}}{8 c^{5}}\)	\(352\)
default	\(\text {Expression too large to display}\)	\(1491\)

Input:

int((b*x^2+a)^(5/2)/x^6/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(b*x^2+a)^(1/2)*(120*a^2*d^4*x^4+280*a*b*c^2*d^2*x^4+184*b^2*c^4*x^ 
4-60*a^2*c*d^3*x^3-135*a*b*c^3*d*x^3+40*a^2*c^2*d^2*x^2+88*a*b*c^4*x^2-30* 
a^2*c^3*d*x+24*a^2*c^4)/c^5/x^5+1/8/c^5*(-8*(a^3*d^6+3*a^2*b*c^2*d^4+3*a*b 
^2*c^4*d^2+b^3*c^6)/c/d/((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))+d*a^(1/2)*(8*a^2*d^4+20*a*b*c^2*d^2+15*b 
^2*c^4)/c*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))
 

Fricas [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 1235, normalized size of antiderivative = 4.35 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)/x^6/(d*x+c),x, algorithm="fricas")
 

Output:

[1/240*(120*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*sqrt(b*c^2 + a*d^2)*x^5*lo 
g((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)) + 
 15*(15*b^2*c^4*d + 20*a*b*c^2*d^3 + 8*a^2*d^5)*sqrt(a)*x^5*log(-(b*x^2 + 
2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(30*a^2*c^4*d*x - 24*a^2*c^5 - 8 
*(23*b^2*c^5 + 35*a*b*c^3*d^2 + 15*a^2*c*d^4)*x^4 + 15*(9*a*b*c^4*d + 4*a^ 
2*c^2*d^3)*x^3 - 8*(11*a*b*c^5 + 5*a^2*c^3*d^2)*x^2)*sqrt(b*x^2 + a))/(c^6 
*x^5), -1/240*(240*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*sqrt(-b*c^2 - a*d^2 
)*x^5*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)) - 15*(15*b^2*c^4*d + 20*a*b*c^2*d^3 + 
 8*a^2*d^5)*sqrt(a)*x^5*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2 
) - 2*(30*a^2*c^4*d*x - 24*a^2*c^5 - 8*(23*b^2*c^5 + 35*a*b*c^3*d^2 + 15*a 
^2*c*d^4)*x^4 + 15*(9*a*b*c^4*d + 4*a^2*c^2*d^3)*x^3 - 8*(11*a*b*c^5 + 5*a 
^2*c^3*d^2)*x^2)*sqrt(b*x^2 + a))/(c^6*x^5), -1/120*(15*(15*b^2*c^4*d + 20 
*a*b*c^2*d^3 + 8*a^2*d^5)*sqrt(-a)*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) 
- 60*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*sqrt(b*c^2 + a*d^2)*x^5*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)) - (30*a^ 
2*c^4*d*x - 24*a^2*c^5 - 8*(23*b^2*c^5 + 35*a*b*c^3*d^2 + 15*a^2*c*d^4)*x^ 
4 + 15*(9*a*b*c^4*d + 4*a^2*c^2*d^3)*x^3 - 8*(11*a*b*c^5 + 5*a^2*c^3*d^...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{6} \left (c + d x\right )}\, dx \] Input:

integrate((b*x**2+a)**(5/2)/x**6/(d*x+c),x)
 

Output:

Integral((a + b*x**2)**(5/2)/(x**6*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/x^6/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x + c)*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (251) = 502\).

Time = 0.15 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.96 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)/x^6/(d*x+c),x, algorithm="giac")
 

Output:

2*(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*arctan(-((sqrt(b 
)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - 
 a*d^2)*c^6) - 1/4*(15*a*b^2*c^4*d + 20*a^2*b*c^2*d^3 + 8*a^3*d^5)*arctan( 
-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*c^6) - 1/60*(135*(sqrt( 
b)*x - sqrt(b*x^2 + a))^9*a*b^2*c^3*d + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9 
*a^2*b*c*d^3 - 360*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(5/2)*c^4 - 360*(sq 
rt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(3/2)*c^2*d^2 - 120*(sqrt(b)*x - sqrt(b 
*x^2 + a))^8*a^3*sqrt(b)*d^4 - 150*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^2*b^2 
*c^3*d - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^3*b*c*d^3 + 720*(sqrt(b)*x 
- sqrt(b*x^2 + a))^6*a^2*b^(5/2)*c^4 + 1200*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
6*a^3*b^(3/2)*c^2*d^2 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*sqrt(b)*d^ 
4 - 1120*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(5/2)*c^4 - 1600*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^4*a^4*b^(3/2)*c^2*d^2 - 720*(sqrt(b)*x - sqrt(b*x^2 + 
a))^4*a^5*sqrt(b)*d^4 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^4*b^2*c^3*d 
+ 120*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^5*b*c*d^3 + 560*(sqrt(b)*x - sqrt( 
b*x^2 + a))^2*a^4*b^(5/2)*c^4 + 1040*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b 
^(3/2)*c^2*d^2 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*sqrt(b)*d^4 - 135 
*(sqrt(b)*x - sqrt(b*x^2 + a))*a^5*b^2*c^3*d - 60*(sqrt(b)*x - sqrt(b*x^2 
+ a))*a^6*b*c*d^3 - 184*a^5*b^(5/2)*c^4 - 280*a^6*b^(3/2)*c^2*d^2 - 120*a^ 
7*sqrt(b)*d^4)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*c^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^6\,\left (c+d\,x\right )} \,d x \] Input:

int((a + b*x^2)^(5/2)/(x^6*(c + d*x)),x)
 

Output:

int((a + b*x^2)^(5/2)/(x^6*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^6 (c+d x)} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{x^{6} \left (d x +c \right )}d x \] Input:

int((b*x^2+a)^(5/2)/x^6/(d*x+c),x)
 

Output:

int((b*x^2+a)^(5/2)/x^6/(d*x+c),x)