3.656 \(\int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx\)

Optimal. Leaf size=334 \[ -5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac{5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{3/2}}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]

[Out]

(-5*(b^3*c^3 - 19*a*b^2*c^2*d - 45*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(64*b*d) - (5*(b^2*c^2 - 18*a*b*c*d - 31*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)
^(3/2))/(96*d) + (5*b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*d) + (5*b
*(a + b*x)^(3/2)*(c + d*x)^(5/2))/4 - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/x - 5*a^
(3/2)*c^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]
)] - (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^
4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(3/2)
)

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Rubi [A]  time = 1.22014, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -5 a^{3/2} c^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{96 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{64 b d}-\frac{5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{3/2}}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x}+\frac{5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b^3*c^3 - 19*a*b^2*c^2*d - 45*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(64*b*d) - (5*(b^2*c^2 - 18*a*b*c*d - 31*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)
^(3/2))/(96*d) + (5*b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*d) + (5*b
*(a + b*x)^(3/2)*(c + d*x)^(5/2))/4 - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/x - 5*a^
(3/2)*c^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]
)] - (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^
4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(3/2)
)

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Rubi in Sympy [A]  time = 173.006, size = 318, normalized size = 0.95 \[ - 5 a^{\frac{3}{2}} c^{\frac{3}{2}} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4} + \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{35 a d}{24} + \frac{5 b c}{24}\right ) - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{x} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a^{2} d^{2} + 14 a b c d + b^{2} c^{2}\right )}{32 b} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a^{3} d^{3} - 19 a^{2} b c d^{2} - 45 a b^{2} c^{2} d - b^{3} c^{3}\right )}{64 b d} - \frac{5 \left (a^{4} d^{4} - 20 a^{3} b c d^{3} - 90 a^{2} b^{2} c^{2} d^{2} - 20 a b^{3} c^{3} d + b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**2,x)

[Out]

-5*a**(3/2)*c**(3/2)*(a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d
*x))) + 5*b*(a + b*x)**(3/2)*(c + d*x)**(5/2)/4 + (a + b*x)**(3/2)*(c + d*x)**(3
/2)*(35*a*d/24 + 5*b*c/24) - (a + b*x)**(5/2)*(c + d*x)**(5/2)/x + 5*(a + b*x)**
(3/2)*sqrt(c + d*x)*(a**2*d**2 + 14*a*b*c*d + b**2*c**2)/(32*b) - 5*sqrt(a + b*x
)*sqrt(c + d*x)*(a**3*d**3 - 19*a**2*b*c*d**2 - 45*a*b**2*c**2*d - b**3*c**3)/(6
4*b*d) - 5*(a**4*d**4 - 20*a**3*b*c*d**3 - 90*a**2*b**2*c**2*d**2 - 20*a*b**3*c*
*3*d + b**4*c**4)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(64*b**(3
/2)*d**(3/2))

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Mathematica [A]  time = 0.325016, size = 318, normalized size = 0.95 \[ \frac{1}{384} \left (960 a^{3/2} c^{3/2} \log (x) (a d+b c)-960 a^{3/2} c^{3/2} (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (15 a^3 d^3 x+a^2 b d \left (-192 c^2+601 c d x+118 d^2 x^2\right )+a b^2 d x \left (601 c^2+452 c d x+136 d^2 x^2\right )+b^3 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{b d x}-\frac{15 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2} d^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3*x + a^2*b*d*(-192*c^2 + 601*c*d*x +
118*d^2*x^2) + a*b^2*d*x*(601*c^2 + 452*c*d*x + 136*d^2*x^2) + b^3*x*(15*c^3 + 1
18*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3)))/(b*d*x) + 960*a^(3/2)*c^(3/2)*(b*c +
a*d)*Log[x] - 960*a^(3/2)*c^(3/2)*(b*c + a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt
[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]] - (15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^
2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sq
rt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(b^(3/2)*d^(3/2)))/384

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Maple [B]  time = 0.028, size = 950, normalized size = 2.8 \[ -{\frac{1}{384\,bdx}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{4}{b}^{3}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-272\,{x}^{3}a{b}^{2}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-272\,{x}^{3}{b}^{3}c{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+15\,{a}^{4}{d}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x\sqrt{ac}-300\,{a}^{3}{d}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cxb\sqrt{ac}-1350\,{a}^{2}{d}^{2}{b}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}x\sqrt{ac}-300\,{b}^{3}{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) axd\sqrt{ac}+15\,{b}^{4}{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x\sqrt{ac}+960\,{a}^{3}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){d}^{2}xb\sqrt{bd}+960\,{a}^{2}{c}^{3}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){b}^{2}xd\sqrt{bd}-236\,{a}^{2}{d}^{3}{x}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}\sqrt{ac}-904\,a{b}^{2}c{d}^{2}{x}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-236\,{b}^{3}{c}^{2}{x}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}\sqrt{ac}-30\,{a}^{3}{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{bd}\sqrt{ac}-1202\,{a}^{2}c{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}xb\sqrt{bd}\sqrt{ac}-1202\,a{b}^{2}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}xd\sqrt{bd}\sqrt{ac}-30\,{b}^{3}{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{bd}\sqrt{ac}+384\,{a}^{2}b{c}^{2}d\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^4*b^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)*(a*c)^(1/2)-272*x^3*a*b^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b
*d)^(1/2)*(a*c)^(1/2)-272*x^3*b^3*c*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1
/2)*(a*c)^(1/2)+15*a^4*d^4*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*(a*c)^(1/2)-300*a^3*d^3*ln(1/2*(2*b*d*x+2*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c*x*b*(a*c)^(1/2)-1
350*a^2*d^2*b^2*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*
d+b*c)/(b*d)^(1/2))*c^2*x*(a*c)^(1/2)-300*b^3*c^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*x*d*(a*c)^(1/2)+15*b^4*c
^4*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^
(1/2))*x*(a*c)^(1/2)+960*a^3*c^2*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)+2*a*c)/x)*d^2*x*b*(b*d)^(1/2)+960*a^2*c^3*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*b^2*x*d*(b*d)^(1/2)-236*a^2*d^3
*x^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*(a*c)^(1/2)-904*a*b^2*c*d^2*x
^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-236*b^3*c^2*x^2*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*d*(b*d)^(1/2)*(a*c)^(1/2)-30*a^3*d^3*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*x*(b*d)^(1/2)*(a*c)^(1/2)-1202*a^2*c*d^2*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*x*b*(b*d)^(1/2)*(a*c)^(1/2)-1202*a*b^2*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*x*d*(b*d)^(1/2)*(a*c)^(1/2)-30*b^3*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*(b*
d)^(1/2)*(a*c)^(1/2)+384*a^2*b*c^2*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)
*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x/d/b/(b*d)^(1/2)/(a*c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.6329, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(960*(a*b^2*c^2*d + a^2*b*c*d^2)*sqrt(a*c)*sqrt(b*d)*x*log((8*a^2*c^2 + (
b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*
x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 15*(b^4*c^4 - 20*a*b^3*c^
3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*x*log(-4*(2*b^2*d^2*x + b^2
*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c
*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)) + 4*(48*b^3*d^3*x^4 - 192*a^2
*b*c^2*d + 136*(b^3*c*d^2 + a*b^2*d^3)*x^3 + 2*(59*b^3*c^2*d + 226*a*b^2*c*d^2 +
59*a^2*b*d^3)*x^2 + (15*b^3*c^3 + 601*a*b^2*c^2*d + 601*a^2*b*c*d^2 + 15*a^3*d^
3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(b*d)*b*d*x), 1/384*(480*(a*b^
2*c^2*d + a^2*b*c*d^2)*sqrt(a*c)*sqrt(-b*d)*x*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*
c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x
+ c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2
*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*
d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) + 2*(48*b^3*d^3*x^4 - 192*a^2*b*c^2*d + 13
6*(b^3*c*d^2 + a*b^2*d^3)*x^3 + 2*(59*b^3*c^2*d + 226*a*b^2*c*d^2 + 59*a^2*b*d^3
)*x^2 + (15*b^3*c^3 + 601*a*b^2*c^2*d + 601*a^2*b*c*d^2 + 15*a^3*d^3)*x)*sqrt(-b
*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*b*d*x), -1/768*(1920*(a*b^2*c^2*d +
a^2*b*c*d^2)*sqrt(-a*c)*sqrt(b*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a
*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^
2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*x*log(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqr
t(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b
^2*c*d + a*b*d^2)*x)*sqrt(b*d)) - 4*(48*b^3*d^3*x^4 - 192*a^2*b*c^2*d + 136*(b^3
*c*d^2 + a*b^2*d^3)*x^3 + 2*(59*b^3*c^2*d + 226*a*b^2*c*d^2 + 59*a^2*b*d^3)*x^2
+ (15*b^3*c^3 + 601*a*b^2*c^2*d + 601*a^2*b*c*d^2 + 15*a^3*d^3)*x)*sqrt(b*d)*sqr
t(b*x + a)*sqrt(d*x + c))/(sqrt(b*d)*b*d*x), -1/384*(960*(a*b^2*c^2*d + a^2*b*c*
d^2)*sqrt(-a*c)*sqrt(-b*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt
(b*x + a)*sqrt(d*x + c))) + 15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 -
20*a^3*b*c*d^3 + a^4*d^4)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*
x + a)*sqrt(d*x + c)*b*d)) - 2*(48*b^3*d^3*x^4 - 192*a^2*b*c^2*d + 136*(b^3*c*d^
2 + a*b^2*d^3)*x^3 + 2*(59*b^3*c^2*d + 226*a*b^2*c*d^2 + 59*a^2*b*d^3)*x^2 + (15
*b^3*c^3 + 601*a*b^2*c^2*d + 601*a^2*b*c*d^2 + 15*a^3*d^3)*x)*sqrt(-b*d)*sqrt(b*
x + a)*sqrt(d*x + c))/(sqrt(-b*d)*b*d*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.739171, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x