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

Optimal. Leaf size=380 \[ -\frac{5 \sqrt{a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{96 c^2 x^2}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{192 a c^2 x}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )}{64 a c^2}+\frac{5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}-\frac{5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{24 c x^3} \]

[Out]

(5*d*(b^3*c^3 + 45*a*b^2*c^2*d + 19*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a*c^2) - (5*(3*b*c + a*d)*(b^2*c^2 + 24*a*b*c*d - a^2*d^2)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(192*a*c^2*x) - (5*(3*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt
[a + b*x]*(c + d*x)^(5/2))/(96*c^2*x^2) - (5*(b*c + a*d)*(a + b*x)^(3/2)*(c + d*
x)^(5/2))/(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(4*x^4) + (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[c
]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(3/2)) + 5*b^(3/2)*d^(3
/2)*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 1.34945, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{5 \sqrt{a+b x} (c+d x)^{5/2} \left (-a^2 d^2+14 a b c d+3 b^2 c^2\right )}{96 c^2 x^2}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c) \left (-a^2 d^2+24 a b c d+b^2 c^2\right )}{192 a c^2 x}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} \left (-a^3 d^3+19 a^2 b c d^2+45 a b^2 c^2 d+b^3 c^3\right )}{64 a c^2}+\frac{5 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{3/2}}+5 b^{3/2} d^{3/2} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{4 x^4}-\frac{5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{24 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

(5*d*(b^3*c^3 + 45*a*b^2*c^2*d + 19*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a*c^2) - (5*(3*b*c + a*d)*(b^2*c^2 + 24*a*b*c*d - a^2*d^2)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(192*a*c^2*x) - (5*(3*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*Sqrt
[a + b*x]*(c + d*x)^(5/2))/(96*c^2*x^2) - (5*(b*c + a*d)*(a + b*x)^(3/2)*(c + d*
x)^(5/2))/(24*c*x^3) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(4*x^4) + (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[c
]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(3/2)) + 5*b^(3/2)*d^(3
/2)*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.34593, size = 364, normalized size = 0.96 \[ \frac{1}{384} \left (-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (136 c^2+452 c d x+601 d^2 x^2\right )+a b^2 c x^2 \left (118 c^2+601 c d x-192 d^2 x^2\right )+15 b^3 c^3 x^3\right )}{a c x^4}-\frac{15 \log (x) \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right )}{a^{3/2} c^{3/2}}+\frac{15 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2} c^{3/2}}+960 b^{3/2} d^{3/2} (a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 962, normalized size = 2.5 \[{\frac{1}{384\,ac{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}\sqrt{bd}-300\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}\sqrt{bd}-1350\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}\sqrt{bd}-300\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d\sqrt{bd}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}\sqrt{bd}+960\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}{a}^{2}{b}^{2}c{d}^{3}\sqrt{ac}+960\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}a{b}^{3}{c}^{2}{d}^{2}\sqrt{ac}+384\,{x}^{4}a{b}^{2}c{d}^{2}\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}-30\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{3}\sqrt{bd}{a}^{3}{x}^{3}\sqrt{ac}-1202\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}b\sqrt{bd}c{a}^{2}{x}^{3}\sqrt{ac}-1202\,\sqrt{d{x}^{2}b+adx+bcx+ac}d{b}^{2}\sqrt{bd}{c}^{2}a{x}^{3}\sqrt{ac}-30\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}\sqrt{bd}{x}^{3}\sqrt{ac}-236\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}c{a}^{3}{x}^{2}\sqrt{ac}-904\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}-236\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}a{x}^{2}\sqrt{ac}-272\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{c}^{2}{a}^{3}x\sqrt{ac}-272\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}{a}^{2}x\sqrt{ac}-96\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c*(15*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)*x^4*a^4*d^4*(b*d)^(1/2)-300*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)*x^4*a^3*b*c*d^3*(b*d)^(1/2
)-1350*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)*x
^4*a^2*b^2*c^2*d^2*(b*d)^(1/2)-300*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)*x^4*a*b^3*c^3*d*(b*d)^(1/2)+15*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)*x^4*b^4*c^4*(b*d)^(1/2)+960*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^4*a^2*b^2*c*d^3*(a*c)^(1/2)+960*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^4*a*b^3*c^2*d^2*(a*c)^(1/2)+384*x^4*a*
b^2*c*d^2*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)-30*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*d^3*(b*d)^(1/2)*a^3*x^3*(a*c)^(1/2)-1202*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*d^2*b*(b*d)^(1/2)*c*a^2*x^3*(a*c)^(1/2)-1202*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*d*b^2*(b*d)^(1/2)*c^2*a*x^3*(a*c)^(1/2)-30*c^3*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*b^3*(b*d)^(1/2)*x^3*(a*c)^(1/2)-236*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^
2*(b*d)^(1/2)*c*a^3*x^2*(a*c)^(1/2)-904*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b*(b*d
)^(1/2)*c^2*a^2*x^2*(a*c)^(1/2)-236*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^2*(b*d
)^(1/2)*a*x^2*(a*c)^(1/2)-272*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*(b*d)^(1/2)*c^2*
a^3*x*(a*c)^(1/2)-272*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*a^2*x*(a
*c)^(1/2)-96*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*(a*c)^(1/2))/(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(a*c)^(1/2)/x^4/(b*d)^(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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.98155, 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^5,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^4*log(8*b^2*d^2*x^
2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x +
a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*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)*x^4*log((4*(2*a^2*c^2 + (a*b*c^2 +
a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^
2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) + 4*(192*a*b^2*c*d^2*x^4 -
48*a^3*c^3 - (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^3
- 2*(59*a*b^2*c^3 + 226*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 - 136*(a^2*b*c^3 + a^3*c
^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c*x^4), 1/768*(1920
*(a*b^2*c^2*d + a^2*b*c*d^2)*sqrt(a*c)*sqrt(-b*d)*x^4*arctan(1/2*(2*b*d*x + b*c
+ a*d)/(sqrt(-b*d)*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^4*log((4*(2*a^2*c^2 + (a*b*c^
2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d
+ a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) + 4*(192*a*b^2*c*d^2*x
^4 - 48*a^3*c^3 - (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^3 - 2*(59*a*b^2*c^3 + 226*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 - 136*(a^2*b*c^3 + a
^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c*x^4), 1/384*(
480*(a*b^2*c^2*d + a^2*b*c*d^2)*sqrt(-a*c)*sqrt(b*d)*x^4*log(8*b^2*d^2*x^2 + b^2
*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqr
t(d*x + c) + 8*(b^2*c*d + a*b*d^2)*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)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqr
t(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) + 2*(192*a*b^2*c*d^2*x^4 - 48*a^3*c^3
- (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^3 - 2*(59*a*b
^2*c^3 + 226*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 - 136*(a^2*b*c^3 + a^3*c^2*d)*x)*sq
rt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a*c*x^4), 1/384*(960*(a*b^2*c^
2*d + a^2*b*c*d^2)*sqrt(-a*c)*sqrt(-b*d)*x^4*arctan(1/2*(2*b*d*x + b*c + a*d)/(s
qrt(-b*d)*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^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*s
qrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) + 2*(192*a*b^2*c*d^2*x^4 - 48*a^3*c
^3 - (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^3 - 2*(59*a
*b^2*c^3 + 226*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 - 136*(a^2*b*c^3 + a^3*c^2*d)*x)*
sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a*c*x^4)]

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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**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.783832, 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^5,x, algorithm="giac")

[Out]

sage0*x