3.652 \(\int \frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=239 \[ -\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^3*x) - ((b*c - a*d)^3*S
qrt[a + b*x]*(c + d*x)^(3/2))/(64*a*c^3*x^2) - ((b*c - a*d)^2*Sqrt[a + b*x]*(c +
d*x)^(5/2))/(16*c^3*x^3) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*c^2
*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*c*x^5) - (3*(b*c - a*d)^5*ArcTanh[(
Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(7/2))

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Rubi [A]  time = 0.47618, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{7/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^3*x) - ((b*c - a*d)^3*S
qrt[a + b*x]*(c + d*x)^(3/2))/(64*a*c^3*x^2) - ((b*c - a*d)^2*Sqrt[a + b*x]*(c +
d*x)^(5/2))/(16*c^3*x^3) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*c^2
*x^4) - ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*c*x^5) - (3*(b*c - a*d)^5*ArcTanh[(
Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(7/2))

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Rubi in Sympy [A]  time = 50.0134, size = 216, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5 c x^{5}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{8 a c x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}}{16 a^{2} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 a^{2} c^{2} x^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4}}{128 a^{2} c^{3} x} + \frac{3 \left (a d - b c\right )^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{5}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**6,x)

[Out]

-(a + b*x)**(5/2)*(c + d*x)**(5/2)/(5*c*x**5) + (a + b*x)**(5/2)*(c + d*x)**(3/2
)*(a*d - b*c)/(8*a*c*x**4) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)**2/(16*a
**2*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**3/(64*a**2*c**2*x**2)
- 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**4/(128*a**2*c**3*x) + 3*(a*d - b*c)
**5*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(128*a**(5/2)*c**(7/2))

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Mathematica [A]  time = 0.364419, size = 270, normalized size = 1.13 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )+2 a^3 b c x \left (168 c^3+256 c^2 d x+23 c d^2 x^2-35 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (124 c^2+233 c d x+64 d^2 x^2\right )+10 a b^3 c^3 x^3 (c+7 d x)-15 b^4 c^4 x^4\right )+15 x^5 \log (x) (b c-a d)^5-15 x^5 (b c-a d)^5 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{1280 a^{5/2} c^{7/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^6,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*b^4*c^4*x^4 + 10*a*b^3*c^3*
x^3*(c + 7*d*x) + 2*a^2*b^2*c^2*x^2*(124*c^2 + 233*c*d*x + 64*d^2*x^2) + 2*a^3*b
*c*x*(168*c^3 + 256*c^2*d*x + 23*c*d^2*x^2 - 35*d^3*x^3) + a^4*(128*c^4 + 176*c^
3*d*x + 8*c^2*d^2*x^2 - 10*c*d^3*x^3 + 15*d^4*x^4)) + 15*(b*c - a*d)^5*x^5*Log[x
] - 15*(b*c - a*d)^5*x^5*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a +
b*x]*Sqrt[c + d*x]])/(1280*a^(5/2)*c^(7/2)*x^5)

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Maple [B]  time = 0.029, size = 967, normalized size = 4.1 \[{\frac{1}{1280\,{a}^{2}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}+150\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}+75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-256\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-92\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-932\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-1024\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-496\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-672\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-256\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^6,x)

[Out]

1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^3*(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^5*a^5*d^5-75*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^5*a^4*b*c*d^4+150*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^5*a^3*b^2*c^2*d^3-15
0*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^5*a^
2*b^3*c^3*d^2+75*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^5*a*b^4*c^4*d-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^5*b^5*c^5-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x
^4*a^4*d^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3-256*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^2*c^2*d^2-140*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d+30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^4*b^4*c^4+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^4
*c*d^3-92*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2-932*(a*c
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d-20*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*x^2*a^4*c^2*d^2-1024*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b
*c^3*d-496*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b^2*c^4-352*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d-672*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x*a^3*b*c^4-256*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^4*(a*c)
^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(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)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.73742, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{5} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (128 \, a^{4} c^{4} -{\left (15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d - 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (5 \, a b^{3} c^{4} + 233 \, a^{2} b^{2} c^{3} d + 23 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (31 \, a^{2} b^{2} c^{4} + 64 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} x^{2} + 16 \,{\left (21 \, a^{3} b c^{4} + 11 \, a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, \sqrt{a c} a^{2} c^{3} x^{5}}, -\frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (128 \, a^{4} c^{4} -{\left (15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d - 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (5 \, a b^{3} c^{4} + 233 \, a^{2} b^{2} c^{3} d + 23 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (31 \, a^{2} b^{2} c^{4} + 64 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} x^{2} + 16 \,{\left (21 \, a^{3} b c^{4} + 11 \, a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, \sqrt{-a c} a^{2} c^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[-1/2560*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3
+ 5*a^4*b*c*d^4 - a^5*d^5)*x^5*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*(128*a^4*c^4 - (15*b^4*c^4 - 70*a*b^3*c^
3*d - 128*a^2*b^2*c^2*d^2 + 70*a^3*b*c*d^3 - 15*a^4*d^4)*x^4 + 2*(5*a*b^3*c^4 +
233*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 5*a^4*c*d^3)*x^3 + 8*(31*a^2*b^2*c^4 + 64
*a^3*b*c^3*d + a^4*c^2*d^2)*x^2 + 16*(21*a^3*b*c^4 + 11*a^4*c^3*d)*x)*sqrt(a*c)*
sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^3*x^5), -1/1280*(15*(b^5*c^5 - 5*a
*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*
x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a
*c)) + 2*(128*a^4*c^4 - (15*b^4*c^4 - 70*a*b^3*c^3*d - 128*a^2*b^2*c^2*d^2 + 70*
a^3*b*c*d^3 - 15*a^4*d^4)*x^4 + 2*(5*a*b^3*c^4 + 233*a^2*b^2*c^3*d + 23*a^3*b*c^
2*d^2 - 5*a^4*c*d^3)*x^3 + 8*(31*a^2*b^2*c^4 + 64*a^3*b*c^3*d + a^4*c^2*d^2)*x^2
+ 16*(21*a^3*b*c^4 + 11*a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(
sqrt(-a*c)*a^2*c^3*x^5)]

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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)**(3/2)/x**6,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError