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

Optimal. Leaf size=201 \[ -\frac{3 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^2 c^2 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{32 a c^2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4} \]

[Out]

(3*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c^2*x) - ((b*c - a*d)^2*Sq
rt[a + b*x]*(c + d*x)^(3/2))/(32*a*c^2*x^2) - ((b*c - a*d)*Sqrt[a + b*x]*(c + d*
x)^(5/2))/(8*c^2*x^3) - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*c*x^4) - (3*(b*c -
a*d)^4*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(
5/2))

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Rubi [A]  time = 0.378667, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^2 c^2 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{32 a c^2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^2*c^2*x) - ((b*c - a*d)^2*Sq
rt[a + b*x]*(c + d*x)^(3/2))/(32*a*c^2*x^2) - ((b*c - a*d)*Sqrt[a + b*x]*(c + d*
x)^(5/2))/(8*c^2*x^3) - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*c*x^4) - (3*(b*c -
a*d)^4*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(
5/2))

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Rubi in Sympy [A]  time = 36.0642, size = 180, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4 c x^{4}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{8 a c x^{3}} + \frac{3 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}}{32 a^{2} c x^{2}} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 a^{2} c^{2} x} - \frac{3 \left (a d - b c\right )^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(3/2)*(c + d*x)**(5/2)/(4*c*x**4) + (a + b*x)**(3/2)*(c + d*x)**(3/2
)*(a*d - b*c)/(8*a*c*x**3) + 3*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2/(32
*a**2*c*x**2) + 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3/(64*a**2*c**2*x) -
3*(a*d - b*c)**4*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(5/
2)*c**(5/2))

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Mathematica [A]  time = 0.259575, size = 215, normalized size = 1.07 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (16 c^3+24 c^2 d x+2 c d^2 x^2-3 d^3 x^3\right )+a^2 b c x \left (24 c^2+44 c d x+11 d^2 x^2\right )+a b^2 c^2 x^2 (2 c+11 d x)-3 b^3 c^3 x^3\right )+3 x^4 \log (x) (b c-a d)^4-3 x^4 (b c-a d)^4 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{128 a^{5/2} c^{5/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*b^3*c^3*x^3 + a*b^2*c^2*x^2*
(2*c + 11*d*x) + a^2*b*c*x*(24*c^2 + 44*c*d*x + 11*d^2*x^2) + a^3*(16*c^3 + 24*c
^2*d*x + 2*c*d^2*x^2 - 3*d^3*x^3)) + 3*(b*c - a*d)^4*x^4*Log[x] - 3*(b*c - a*d)^
4*x^4*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]]
)/(128*a^(5/2)*c^(5/2)*x^4)

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Maple [B]  time = 0.025, size = 705, normalized size = 3.5 \[ -{\frac{1}{128\,{a}^{2}{c}^{2}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}+22\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}+22\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d-6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}+88\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}+48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d+48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}+32\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2*(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)*x^4*a^4*d^4-12*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+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)
*x^4*b^4*c^4-6*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^3+22*(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+22*(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-6*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*x^3*b^3*c^3+4*(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+88*(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+4*(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+48*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x*a^3*c^2*d+48*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c^3+32*(
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)*(d*x + c)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27572, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (16 \, a^{3} c^{3} -{\left (3 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (a b^{2} c^{3} + 22 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} + 24 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{256 \, \sqrt{a c} a^{2} c^{2} x^{4}}, -\frac{3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (16 \, a^{3} c^{3} -{\left (3 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d - 11 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (a b^{2} c^{3} + 22 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} + 24 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{128 \, \sqrt{-a c} a^{2} c^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*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*(16*a^3*c^3 - (3*b^3*c^3 - 11*a*b^2*c^2*d - 11*a^2*b*c*d^2 + 3*a
^3*d^3)*x^3 + 2*(a*b^2*c^3 + 22*a^2*b*c^2*d + a^3*c*d^2)*x^2 + 24*(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^2*x^4), -1/
128*(3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x
^4*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*(16*a^3*c^3 - (3*b^3*c^3 - 11*a*b^2*c^2*d - 11*a^2*b*c*d^2 + 3*a^3*d^3)*
x^3 + 2*(a*b^2*c^3 + 22*a^2*b*c^2*d + a^3*c*d^2)*x^2 + 24*(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^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**5, x)

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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)*(d*x + c)^(3/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError