3.657 \(\int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=319 \[ \frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )}{12 c}+\frac{5}{8} \sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac{5 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac{5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{12 c}-\frac{5}{4} \sqrt{a} \sqrt{c} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) \]

[Out]

(5*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/8 + (5*(b^2*c
^2 + 8*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*c) + (5*b*(5*b*c
+ 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(12*c) - (5*(b*c + a*d)*(a + b*x)^(3/2)*
(c + d*x)^(5/2))/(4*c*x) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(2*x^2) - (5*Sqrt[a
]*Sqrt[c]*(3*b*c + a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/4 + (5*(b*c + a*d)*(b^2*c^2 + 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqr
t[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*Sqrt[d])

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Rubi [A]  time = 1.21164, antiderivative size = 319, 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 \[ \frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )}{12 c}+\frac{5}{8} \sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac{5 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac{5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x}+\frac{5 b \sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{12 c}-\frac{5}{4} \sqrt{a} \sqrt{c} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^3,x]

[Out]

(5*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/8 + (5*(b^2*c
^2 + 8*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*c) + (5*b*(5*b*c
+ 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(12*c) - (5*(b*c + a*d)*(a + b*x)^(3/2)*
(c + d*x)^(5/2))/(4*c*x) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(2*x^2) - (5*Sqrt[a
]*Sqrt[c]*(3*b*c + a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/4 + (5*(b*c + a*d)*(b^2*c^2 + 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqr
t[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*Sqrt[d])

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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**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.26871, size = 309, normalized size = 0.97 \[ \frac{1}{48} \left (\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (-3 a^2 \left (4 c^2+18 c d x-11 d^2 x^2\right )+2 a b x \left (-27 c^2+61 c d x+13 d^2 x^2\right )+b^2 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{x^2}+30 \sqrt{a} \sqrt{c} \log (x) \left (3 a^2 d^2+10 a b c d+3 b^2 c^2\right )-30 \sqrt{a} \sqrt{c} \left (3 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{15 \left (a^3 d^3+15 a^2 b c d^2+15 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} \sqrt{d}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^3,x]

[Out]

((2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^2*(4*c^2 + 18*c*d*x - 11*d^2*x^2) + b^2*x^
2*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + 2*a*b*x*(-27*c^2 + 61*c*d*x + 13*d^2*x^2)))/
x^2 + 30*Sqrt[a]*Sqrt[c]*(3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Log[x] - 30*Sqrt[a
]*Sqrt[c]*(3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]] + (15*(b^3*c^3 + 15*a*b^2*c^2*d + 15*a
^2*b*c*d^2 + a^3*d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*
Sqrt[c + d*x]])/(Sqrt[b]*Sqrt[d]))/48

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Maple [B]  time = 0.025, size = 850, normalized size = 2.7 \[{\frac{1}{48\,{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{4}{b}^{2}{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}{x}^{2}\sqrt{ac}+225\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}bc{x}^{2}\sqrt{ac}+225\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ad{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}{x}^{2}\sqrt{ac}-90\,{a}^{3}c\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){d}^{2}{x}^{2}\sqrt{bd}-300\,{a}^{2}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) bd{x}^{2}\sqrt{bd}-90\,a{c}^{3}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){b}^{2}{x}^{2}\sqrt{bd}+52\,{x}^{3}ab{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+52\,{x}^{3}{b}^{2}cd\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+66\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}{a}^{2}\sqrt{ac}{x}^{2}+244\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}a\sqrt{ac}{x}^{2}c+66\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}\sqrt{ac}{x}^{2}-108\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{a}^{2}\sqrt{ac}xc-108\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}a\sqrt{ac}x-24\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^3,x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*x^4*b^2*d^2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*(b*d)^(1/2)+15*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^3*d^3*x^2*(a*c)^(1/2)+225*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^2*d^2*b*c*x^2*
(a*c)^(1/2)+225*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*d*b^2*c^2*x^2*(a*c)^(1/2)+15*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))*b^3*c^3*x^2*(a*c)^(1/2)-9
0*a^3*c*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^2*(b*d)^(1/2)-300*a^2*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)*b*d*x^2*(b*d)^(1/2)-90*a*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^2*(b*d)^(1/2)+52*x^3*a*b*d^2*
(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+52*x^3*b^2*c*d*(b*d)^(1/
2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+66*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*d^2*(b*d)^(1/2)*a^2*(a*c)^(1/2)*x^2+244*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b*(b
*d)^(1/2)*a*(a*c)^(1/2)*x^2*c+66*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^2*(b*d)^(
1/2)*(a*c)^(1/2)*x^2-108*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*(b*d)^(1/2)*a^2*(a*c)
^(1/2)*x*c-108*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*a*(a*c)^(1/2)*x
-24*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*(a*c)^(1/2))/(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)/x^2/(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.0175, 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^3,x, algorithm="fricas")

[Out]

[1/96*(30*(3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sqrt(b*d)*x^2*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^3*c^3 + 15
*a*b^2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^2*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*(8*b^2*d^2*x^4 - 12*a^2*c^2 + 26*(b^
2*c*d + a*b*d^2)*x^3 + (33*b^2*c^2 + 122*a*b*c*d + 33*a^2*d^2)*x^2 - 54*(a*b*c^2
+ a^2*c*d)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(b*d)*x^2), 1/48*(15*
(3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sqrt(-b*d)*x^2*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^3*c^3 + 15*a*b^2*c^
2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^2*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*(8*b^2*d^2*x^4 - 12*a^2*c^2 + 26*(b^2*c*d
+ a*b*d^2)*x^3 + (33*b^2*c^2 + 122*a*b*c*d + 33*a^2*d^2)*x^2 - 54*(a*b*c^2 + a^
2*c*d)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*x^2), -1/96*(60*(3
*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*sqrt(b*d)*x^2*arctan(1/2*(2*a*c +
(b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 15*(b^3*c^3 + 15*a*b^
2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^2*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*(8*b^2*d^2*x^4 - 12*a^2*c^2 + 26*(b^2*c*d
+ a*b*d^2)*x^3 + (33*b^2*c^2 + 122*a*b*c*d + 33*a^2*d^2)*x^2 - 54*(a*b*c^2 + a^
2*c*d)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(b*d)*x^2), -1/48*(30*(3*b
^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*sqrt(-b*d)*x^2*arctan(1/2*(2*a*c + (
b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 15*(b^3*c^3 + 15*a*b^2
*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^2*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*(8*b^2*d^2*x^4 - 12*a^2*c^2 + 26*(b^2*
c*d + a*b*d^2)*x^3 + (33*b^2*c^2 + 122*a*b*c*d + 33*a^2*d^2)*x^2 - 54*(a*b*c^2 +
a^2*c*d)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*x^2)]

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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**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.757718, 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^3,x, algorithm="giac")

[Out]

sage0*x