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

Optimal. Leaf size=233 \[ -\frac{(5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)^2}{64 a^2 c^3 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+3 b c) (b c-a d)}{32 a c^3 x^2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 a c^2 x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4} \]

[Out]

((b*c - a*d)^2*(3*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c^3*x) + ((b
*c - a*d)*(3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*a*c^3*x^2) + ((3*b*
c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*a*c^2*x^3) - ((a + b*x)^(5/2)*(c
+ d*x)^(3/2))/(4*a*c*x^4) - ((b*c - a*d)^3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(7/2))

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Rubi [A]  time = 0.408666, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(5 a d+3 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+3 b c) (b c-a d)^2}{64 a^2 c^3 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+3 b c) (b c-a d)}{32 a c^3 x^2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 a c^2 x^3}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 a c x^4} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(3*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c^3*x) + ((b
*c - a*d)*(3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*a*c^3*x^2) + ((3*b*
c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*a*c^2*x^3) - ((a + b*x)^(5/2)*(c
+ d*x)^(3/2))/(4*a*c*x^4) - ((b*c - a*d)^3*(3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(7/2))

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Rubi in Sympy [A]  time = 37.1781, size = 212, normalized size = 0.91 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4 a c x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (5 a d + 3 b c\right )}{24 a^{2} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a d + 3 b c\right )}{96 a^{2} c^{2} x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a d + 3 b c\right )}{64 a^{2} c^{3} x} + \frac{\left (a d - b c\right )^{3} \left (5 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{5}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(1/2)/x**5,x)

[Out]

-(a + b*x)**(5/2)*(c + d*x)**(3/2)/(4*a*c*x**4) + (a + b*x)**(5/2)*sqrt(c + d*x)
*(5*a*d + 3*b*c)/(24*a**2*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)*(
5*a*d + 3*b*c)/(96*a**2*c**2*x**2) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*
(5*a*d + 3*b*c)/(64*a**2*c**3*x) + (a*d - b*c)**3*(5*a*d + 3*b*c)*atanh(sqrt(c)*
sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(5/2)*c**(7/2))

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Mathematica [A]  time = 0.253805, size = 234, normalized size = 1. \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (72 c^2+20 c d x-31 d^2 x^2\right )+3 a b^2 c^2 x^2 (2 c+3 d x)-9 b^3 c^3 x^3\right )+3 x^4 \log (x) (b c-a d)^3 (5 a d+3 b c)-3 x^4 (b c-a d)^3 (5 a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{384 a^{5/2} c^{7/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^
2*(2*c + 3*d*x) + a^2*b*c*x*(72*c^2 + 20*c*d*x - 31*d^2*x^2) + a^3*(48*c^3 + 8*c
^2*d*x - 10*c*d^2*x^2 + 15*d^3*x^3)) + 3*(b*c - a*d)^3*(3*b*c + 5*a*d)*x^4*Log[x
] - 3*(b*c - a*d)^3*(3*b*c + 5*a*d)*x^4*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sq
rt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(384*a^(5/2)*c^(7/2)*x^4)

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Maple [B]  time = 0.025, size = 705, normalized size = 3. \[{\frac{1}{384\,{a}^{2}{c}^{3}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-36\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}+62\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}-18\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d+18\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-40\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d-12\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-144\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^5,x)

[Out]

1/384*(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^4*a^4*d^4-36*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+18*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+12*l
n((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-9*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^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^3+62*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b*c*d^2-18*(a*c)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^2*c^2*d+18*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x^3*b^3*c^3+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*c*d^2-40
*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*c^2*d-12*(a*c)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c^3-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*x*a^3*c^2*d-144*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c^3
-96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)/x^4/(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)^(3/2)*sqrt(d*x + c)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.714509, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} x^{4} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} c^{3} -{\left (9 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 31 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (9 \, a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{2} c^{3} x^{4}}, -\frac{3 \,{\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (48 \, a^{3} c^{3} -{\left (9 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 31 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (3 \, a b^{2} c^{3} + 10 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (9 \, a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{2} c^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/768*(3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 5*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*(48*a^3*c^3 - (9*b^3*c^3 - 9*a*b^2*c^2*d + 31*a^2*b*c*d^2 -
15*a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 10*a^2*b*c^2*d - 5*a^3*c*d^2)*x^2 + 8*(9*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^2*c^3
*x^4), -1/384*(3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3
- 5*a^4*d^4)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*s
qrt(d*x + c)*a*c)) + 2*(48*a^3*c^3 - (9*b^3*c^3 - 9*a*b^2*c^2*d + 31*a^2*b*c*d^2
- 15*a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 10*a^2*b*c^2*d - 5*a^3*c*d^2)*x^2 + 8*(9*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^2
*c^3*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)**(3/2)*(d*x+c)**(1/2)/x**5,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)^(3/2)*sqrt(d*x + c)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError