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

Optimal. Leaf size=273 \[ \frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

[Out]

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

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Rubi [A]  time = 0.522907, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 (a d+b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{128 a^3 c^3 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c) (b c-a d)^2}{64 a^2 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+b c) (b c-a d)}{16 a c^3 x^3}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 a c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.3987, size = 245, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5 a c x^{5}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d + b c\right )}{8 a^{2} c x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (a d + b c\right )}{16 a^{3} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a d + b c\right )}{64 a^{3} c^{2} x^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (a d + b c\right )}{128 a^{3} c^{3} x} + \frac{3 \left (a d - b c\right )^{4} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{7}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.336899, size = 284, normalized size = 1.04 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )+2 a^3 b c x \left (88 c^3+136 c^2 d x+13 c d^2 x^2-20 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (4 c^2+13 c d x+9 d^2 x^2\right )-10 a b^3 c^3 x^3 (c+4 d x)+15 b^4 c^4 x^4\right )-15 x^5 \log (x) (b c-a d)^4 (a d+b c)+15 x^5 (b c-a d)^4 (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{1280 a^{7/2} c^{7/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(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 + 4*d*x) + 2*a^2*b^2*c^2*x^2*(4*c^2 + 13*c*d*x + 9*d^2*x^2) + 2*a^3*b*c*x*
(88*c^3 + 136*c^2*d*x + 13*c*d^2*x^2 - 20*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)^4*(b*c + a*d)*x^5
*Log[x] + 15*(b*c - a*d)^4*(b*c + a*d)*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^(7/2)*c^(7/2)*x^5)

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Maple [B]  time = 0.028, size = 967, normalized size = 3.5 \[{\frac{1}{1280\,{a}^{3}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-52\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-52\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-544\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-256\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/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^5-45*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+30*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+30*
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^2-45*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+80*(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^3-36*(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+80*(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^
3-52*(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^2-52*(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^2-544*(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-
16*(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-352*(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)^(3/2)*(d*x + c)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.73585, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{5} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (128 \, a^{4} c^{4} +{\left (15 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (5 \, a b^{3} c^{4} - 13 \, a^{2} b^{2} c^{3} d - 13 \, a^{3} b c^{2} d^{2} + 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (a^{2} b^{2} c^{4} + 34 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} x^{2} + 176 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, \sqrt{a c} a^{3} c^{3} x^{5}}, \frac{15 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (128 \, a^{4} c^{4} +{\left (15 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (5 \, a b^{3} c^{4} - 13 \, a^{2} b^{2} c^{3} d - 13 \, a^{3} b c^{2} d^{2} + 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (a^{2} b^{2} c^{4} + 34 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} x^{2} + 176 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, \sqrt{-a c} a^{3} c^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3
*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 - 40*a*b^3*c^3*d
+ 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^4*d^4)*x^4 - 2*(5*a*b^3*c^4 - 13*a
^2*b^2*c^3*d - 13*a^3*b*c^2*d^2 + 5*a^4*c*d^3)*x^3 + 8*(a^2*b^2*c^4 + 34*a^3*b*c
^3*d + a^4*c^2*d^2)*x^2 + 176*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(a*c)*sqrt(b*x + a)
*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^3*x^5), 1/1280*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2
*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*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 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*
a^4*d^4)*x^4 - 2*(5*a*b^3*c^4 - 13*a^2*b^2*c^3*d - 13*a^3*b*c^2*d^2 + 5*a^4*c*d^
3)*x^3 + 8*(a^2*b^2*c^4 + 34*a^3*b*c^3*d + a^4*c^2*d^2)*x^2 + 176*(a^3*b*c^4 + a
^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*c^3*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**6, 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^6,x, algorithm="giac")

[Out]

Exception raised: TypeError