3.395 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{5/2}}+\frac{b^2 \sqrt{a+b x} (3 A b-8 a B)}{64 a^2 x}+\frac{(a+b x)^{3/2} (3 A b-8 a B)}{24 a x^3}+\frac{b \sqrt{a+b x} (3 A b-8 a B)}{32 a x^2}-\frac{A (a+b x)^{5/2}}{4 a x^4} \]

[Out]

(b*(3*A*b - 8*a*B)*Sqrt[a + b*x])/(32*a*x^2) + (b^2*(3*A*b - 8*a*B)*Sqrt[a + b*x
])/(64*a^2*x) + ((3*A*b - 8*a*B)*(a + b*x)^(3/2))/(24*a*x^3) - (A*(a + b*x)^(5/2
))/(4*a*x^4) - (b^3*(3*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(5/2))

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Rubi [A]  time = 0.193849, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{5/2}}+\frac{b^2 \sqrt{a+b x} (3 A b-8 a B)}{64 a^2 x}+\frac{(a+b x)^{3/2} (3 A b-8 a B)}{24 a x^3}+\frac{b \sqrt{a+b x} (3 A b-8 a B)}{32 a x^2}-\frac{A (a+b x)^{5/2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*(3*A*b - 8*a*B)*Sqrt[a + b*x])/(32*a*x^2) + (b^2*(3*A*b - 8*a*B)*Sqrt[a + b*x
])/(64*a^2*x) + ((3*A*b - 8*a*B)*(a + b*x)^(3/2))/(24*a*x^3) - (A*(a + b*x)^(5/2
))/(4*a*x^4) - (b^3*(3*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(5/2))

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Rubi in Sympy [A]  time = 16.7178, size = 133, normalized size = 0.91 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{4 a x^{4}} + \frac{b \sqrt{a + b x} \left (3 A b - 8 B a\right )}{32 a x^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (3 A b - 8 B a\right )}{24 a x^{3}} + \frac{b^{2} \sqrt{a + b x} \left (3 A b - 8 B a\right )}{64 a^{2} x} - \frac{b^{3} \left (3 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x)**(5/2)/(4*a*x**4) + b*sqrt(a + b*x)*(3*A*b - 8*B*a)/(32*a*x**2) + (
a + b*x)**(3/2)*(3*A*b - 8*B*a)/(24*a*x**3) + b**2*sqrt(a + b*x)*(3*A*b - 8*B*a)
/(64*a**2*x) - b**3*(3*A*b - 8*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(64*a**(5/2))

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Mathematica [A]  time = 0.183218, size = 110, normalized size = 0.75 \[ \frac{b^3 (8 a B-3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{5/2}}-\frac{\sqrt{a+b x} \left (16 a^3 (3 A+4 B x)+8 a^2 b x (9 A+14 B x)+6 a b^2 x^2 (A+4 B x)-9 A b^3 x^3\right )}{192 a^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*(-9*A*b^3*x^3 + 6*a*b^2*x^2*(A + 4*B*x) + 16*a^3*(3*A + 4*B*x) +
8*a^2*b*x*(9*A + 14*B*x)))/(192*a^2*x^4) + (b^3*(-3*A*b + 8*a*B)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/(64*a^(5/2))

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Maple [A]  time = 0.019, size = 119, normalized size = 0.8 \[ 2\,{b}^{3} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ({\frac{ \left ( 3\,Ab-8\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,{a}^{2}}}-{\frac{ \left ( 33\,Ab+40\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{384\,a}}+ \left ( -{\frac{11\,Ab}{128}}+{\frac{11\,Ba}{48}} \right ) \left ( bx+a \right ) ^{3/2}+{\frac{a \left ( 3\,Ab-8\,Ba \right ) \sqrt{bx+a}}{128}} \right ) }-{\frac{3\,Ab-8\,Ba}{128\,{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^3*((1/128*(3*A*b-8*B*a)/a^2*(b*x+a)^(7/2)-1/384*(33*A*b+40*B*a)/a*(b*x+a)^(5
/2)+(-11/128*A*b+11/48*B*a)*(b*x+a)^(3/2)+1/128*a*(3*A*b-8*B*a)*(b*x+a)^(1/2))/x
^4/b^4-1/128*(3*A*b-8*B*a)/a^(5/2)*arctanh((b*x+a)^(1/2)/a^(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)*(b*x + a)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225498, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{3} + 3 \,{\left (8 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (56 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + 9 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{384 \, a^{\frac{5}{2}} x^{4}}, -\frac{3 \,{\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (48 \, A a^{3} + 3 \,{\left (8 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \,{\left (56 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + 9 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{192 \, \sqrt{-a} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(3*(8*B*a*b^3 - 3*A*b^4)*x^4*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*
a)/x) + 2*(48*A*a^3 + 3*(8*B*a*b^2 - 3*A*b^3)*x^3 + 2*(56*B*a^2*b + 3*A*a*b^2)*x
^2 + 8*(8*B*a^3 + 9*A*a^2*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(5/2)*x^4), -1/192*(3*
(8*B*a*b^3 - 3*A*b^4)*x^4*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (48*A*a^3 + 3*(8*
B*a*b^2 - 3*A*b^3)*x^3 + 2*(56*B*a^2*b + 3*A*a*b^2)*x^2 + 8*(8*B*a^3 + 9*A*a^2*b
)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^4)]

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Sympy [A]  time = 132.397, size = 1278, normalized size = 8.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-558*A*a**5*b**4*sqrt(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)
**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 1022*A*a**4*b**4*(a + b*
x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a +
b*x)**3 + 384*a**4*(a + b*x)**4) - 770*A*a**3*b**4*(a + b*x)**(5/2)/(-1152*a**8
- 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a
+ b*x)**4) - 132*A*a**3*b**4*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a
+ b*x)**2 + 48*a**3*(a + b*x)**3) + 210*A*a**2*b**4*(a + b*x)**(7/2)/(-1152*a**
8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(
a + b*x)**4) + 160*A*a**2*b**4*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 144*a*
*4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) + 35*A*a**2*b**4*sqrt(a**(-9))*log(-a**5
*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*A*a**2*b**4*sqrt(a**(-9))*log(a**5*sqrt
(a**(-9)) + sqrt(a + b*x))/128 - 60*A*a*b**4*(a + b*x)**(5/2)/(96*a**6 + 144*a**
5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 10*A*a*b**4*sqrt(a + b*x
)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 5*A*a*b**4*sqrt(a**(-7))*log(-
a**4*sqrt(a**(-7)) + sqrt(a + b*x))/8 + 5*A*a*b**4*sqrt(a**(-7))*log(a**4*sqrt(a
**(-7)) + sqrt(a + b*x))/8 + 6*A*b**4*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x +
8*a**2*(a + b*x)**2) + 3*A*b**4*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a +
b*x))/8 - 3*A*b**4*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 66
*B*a**4*b**3*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*
a**3*(a + b*x)**3) + 80*B*a**3*b**3*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 1
44*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 30*B*a**2*b**3*(a + b*x)**(5/2)/(
96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 20*B*a*
*2*b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 5*B*a**2*b
**3*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*B*a**2*b**3*sq
rt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 12*B*a*b**3*(a + b*x)**
(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*B*a*b**3*sqrt(a**(-5))*l
og(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/4 - 3*B*a*b**3*sqrt(a**(-5))*log(a**3*sq
rt(a**(-5)) + sqrt(a + b*x))/4 - B*b**3*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) +
sqrt(a + b*x))/2 + B*b**3*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/
2 - B*b**2*sqrt(a + b*x)/(a*x)

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GIAC/XCAS [A]  time = 0.232803, size = 238, normalized size = 1.63 \[ -\frac{\frac{3 \,{\left (8 \, B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{24 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{4} + 40 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{4} - 88 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{4} + 24 \, \sqrt{b x + a} B a^{4} b^{4} - 9 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{5} + 33 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{5} + 33 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 9 \, \sqrt{b x + a} A a^{3} b^{5}}{a^{2} b^{4} x^{4}}}{192 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(3*(8*B*a*b^4 - 3*A*b^5)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) +
(24*(b*x + a)^(7/2)*B*a*b^4 + 40*(b*x + a)^(5/2)*B*a^2*b^4 - 88*(b*x + a)^(3/2)*
B*a^3*b^4 + 24*sqrt(b*x + a)*B*a^4*b^4 - 9*(b*x + a)^(7/2)*A*b^5 + 33*(b*x + a)^
(5/2)*A*a*b^5 + 33*(b*x + a)^(3/2)*A*a^2*b^5 - 9*sqrt(b*x + a)*A*a^3*b^5)/(a^2*b
^4*x^4))/b