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

Optimal. Leaf size=242 \[ \frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^3 c^2 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a^2 c^2 x^2}-\frac{3 \sqrt{a+b x} (c+d x)^{7/2} (b c-a d)}{40 c^2 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{80 a c^2 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5} \]

[Out]

(-3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^3*c^2*x) + ((b*c - a*d)^3*
Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*a^2*c^2*x^2) - ((b*c - a*d)^2*Sqrt[a + b*x]*(
c + d*x)^(5/2))/(80*a*c^2*x^3) - (3*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(
40*c^2*x^4) - ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*c*x^5) + (3*(b*c - a*d)^5*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(5/2))

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Rubi [A]  time = 0.501622, antiderivative size = 242, 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 \[ \frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^3 c^2 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a^2 c^2 x^2}-\frac{3 \sqrt{a+b x} (c+d x)^{7/2} (b c-a d)}{40 c^2 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{80 a c^2 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^3*c^2*x) + ((b*c - a*d)^3*
Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*a^2*c^2*x^2) - ((b*c - a*d)^2*Sqrt[a + b*x]*(
c + d*x)^(5/2))/(80*a*c^2*x^3) - (3*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(
40*c^2*x^4) - ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*c*x^5) + (3*(b*c - a*d)^5*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(5/2))

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Rubi in Sympy [A]  time = 49.3442, size = 219, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}}}{5 c x^{5}} + \frac{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{40 a c x^{4}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{16 a^{2} c x^{3}} + \frac{3 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}}{64 a^{2} c^{2} x^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4}}{128 a^{3} c^{2} x} - \frac{3 \left (a d - b c\right )^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{7}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(3/2)*(c + d*x)**(7/2)/(5*c*x**5) + 3*(a + b*x)**(3/2)*(c + d*x)**(5
/2)*(a*d - b*c)/(40*a*c*x**4) + (a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)**2
/(16*a**2*c*x**3) + 3*sqrt(a + b*x)*(c + d*x)**(3/2)*(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**3*c**2*x) - 3*(a*
d - b*c)**5*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(128*a**(7/2)*c
**(5/2))

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Mathematica [A]  time = 0.345567, size = 270, normalized size = 1.12 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )+2 a^3 b c x \left (88 c^3+256 c^2 d x+233 c d^2 x^2+35 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (4 c^2+23 c d x+64 d^2 x^2\right )-10 a b^3 c^3 x^3 (c+7 d x)+15 b^4 c^4 x^4\right )-15 x^5 \log (x) (b c-a d)^5+15 x^5 (b c-a d)^5 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{1280 a^{7/2} c^{5/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/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*(4*c^2 + 23*c*d*x + 64*d^2*x^2) + 2*a^3*b*c*x
*(88*c^3 + 256*c^2*d*x + 233*c*d^2*x^2 + 35*d^3*x^3) + a^4*(128*c^4 + 336*c^3*d*
x + 248*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^(7/2)*c^(5/2)*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2*(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^3-1
50*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+932*(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+92*(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+496*(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+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+672*(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)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.8267, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (128 \, a^{4} c^{4} +{\left (15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d + 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (5 \, a b^{3} c^{4} - 23 \, a^{2} b^{2} c^{3} d - 233 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (a^{2} b^{2} c^{4} + 64 \, a^{3} b c^{3} d + 31 \, a^{4} c^{2} d^{2}\right )} x^{2} + 16 \,{\left (11 \, a^{3} b c^{4} + 21 \, a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, \sqrt{a c} a^{3} c^{2} x^{5}}, \frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (128 \, a^{4} c^{4} +{\left (15 \, b^{4} c^{4} - 70 \, a b^{3} c^{3} d + 128 \, a^{2} b^{2} c^{2} d^{2} + 70 \, a^{3} b c d^{3} - 15 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (5 \, a b^{3} c^{4} - 23 \, a^{2} b^{2} c^{3} d - 233 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (a^{2} b^{2} c^{4} + 64 \, a^{3} b c^{3} d + 31 \, a^{4} c^{2} d^{2}\right )} x^{2} + 16 \,{\left (11 \, a^{3} b c^{4} + 21 \, a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, \sqrt{-a c} a^{3} c^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*(d*x + c)^(5/2)/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 -
23*a^2*b^2*c^3*d - 233*a^3*b*c^2*d^2 - 5*a^4*c*d^3)*x^3 + 8*(a^2*b^2*c^4 + 64*a
^3*b*c^3*d + 31*a^4*c^2*d^2)*x^2 + 16*(11*a^3*b*c^4 + 21*a^4*c^3*d)*x)*sqrt(a*c)
*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^2*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 - 23*a^2*b^2*c^3*d - 233*a^3*b*c^
2*d^2 - 5*a^4*c*d^3)*x^3 + 8*(a^2*b^2*c^4 + 64*a^3*b*c^3*d + 31*a^4*c^2*d^2)*x^2
+ 16*(11*a^3*b*c^4 + 21*a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(
sqrt(-a*c)*a^3*c^2*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)**(3/2)*(d*x+c)**(5/2)/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)^(3/2)*(d*x + c)^(5/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError