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

Optimal. Leaf size=142 \[ \frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{3/2}}+\frac{5 b^2 \sqrt{a+b x} (A b-8 a B)}{64 a x}+\frac{(a+b x)^{5/2} (A b-8 a B)}{24 a x^3}+\frac{5 b (a+b x)^{3/2} (A b-8 a B)}{96 a x^2}-\frac{A (a+b x)^{7/2}}{4 a x^4} \]

[Out]

(5*b^2*(A*b - 8*a*B)*Sqrt[a + b*x])/(64*a*x) + (5*b*(A*b - 8*a*B)*(a + b*x)^(3/2
))/(96*a*x^2) + ((A*b - 8*a*B)*(a + b*x)^(5/2))/(24*a*x^3) - (A*(a + b*x)^(7/2))
/(4*a*x^4) + (5*b^3*(A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(3/2))

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Rubi [A]  time = 0.186407, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{3/2}}+\frac{5 b^2 \sqrt{a+b x} (A b-8 a B)}{64 a x}+\frac{(a+b x)^{5/2} (A b-8 a B)}{24 a x^3}+\frac{5 b (a+b x)^{3/2} (A b-8 a B)}{96 a x^2}-\frac{A (a+b x)^{7/2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

(5*b^2*(A*b - 8*a*B)*Sqrt[a + b*x])/(64*a*x) + (5*b*(A*b - 8*a*B)*(a + b*x)^(3/2
))/(96*a*x^2) + ((A*b - 8*a*B)*(a + b*x)^(5/2))/(24*a*x^3) - (A*(a + b*x)^(7/2))
/(4*a*x^4) + (5*b^3*(A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(3/2))

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Rubi in Sympy [A]  time = 16.3237, size = 129, normalized size = 0.91 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{4 a x^{4}} + \frac{5 b^{2} \sqrt{a + b x} \left (A b - 8 B a\right )}{64 a x} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \left (A b - 8 B a\right )}{96 a x^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (A b - 8 B a\right )}{24 a x^{3}} + \frac{5 b^{3} \left (A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x)**(7/2)/(4*a*x**4) + 5*b**2*sqrt(a + b*x)*(A*b - 8*B*a)/(64*a*x) + 5
*b*(a + b*x)**(3/2)*(A*b - 8*B*a)/(96*a*x**2) + (a + b*x)**(5/2)*(A*b - 8*B*a)/(
24*a*x**3) + 5*b**3*(A*b - 8*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(64*a**(3/2))

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Mathematica [A]  time = 0.186023, size = 111, normalized size = 0.78 \[ \frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{3/2}}-\frac{\sqrt{a+b x} \left (16 a^3 (3 A+4 B x)+8 a^2 b x (17 A+26 B x)+2 a b^2 x^2 (59 A+132 B x)+15 A b^3 x^3\right )}{192 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*(15*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 8*a^2*b*x*(17*A + 26*B*x)
+ 2*a*b^2*x^2*(59*A + 132*B*x)))/(192*a*x^4) + (5*b^3*(A*b - 8*a*B)*ArcTanh[Sqr
t[a + b*x]/Sqrt[a]])/(64*a^(3/2))

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Maple [A]  time = 0.02, size = 118, normalized size = 0.8 \[ 2\,{b}^{3} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ( -{\frac{ \left ( 5\,Ab+88\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,a}}+ \left ({\frac{73\,Ba}{48}}-{\frac{73\,Ab}{384}} \right ) \left ( bx+a \right ) ^{5/2}+{\frac{55\,a \left ( Ab-8\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384}}+ \left ({\frac{5\,B{a}^{3}}{16}}-{\frac{5\,A{a}^{2}b}{128}} \right ) \sqrt{bx+a} \right ) }+{\frac{5\,Ab-40\,Ba}{128\,{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^3*((-1/128*(5*A*b+88*B*a)/a*(b*x+a)^(7/2)+(73/48*B*a-73/384*A*b)*(b*x+a)^(5/
2)+55/384*a*(A*b-8*B*a)*(b*x+a)^(3/2)+(5/16*B*a^3-5/128*A*a^2*b)*(b*x+a)^(1/2))/
x^4/b^4+5/128*(A*b-8*B*a)/a^(3/2)*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)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219428, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (8 \, B a b^{3} - A b^{4}\right )} x^{4} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{3} + 3 \,{\left (88 \, B a b^{2} + 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (104 \, B a^{2} b + 59 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + 17 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{384 \, a^{\frac{3}{2}} x^{4}}, \frac{15 \,{\left (8 \, B a b^{3} - A b^{4}\right )} x^{4} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (48 \, A a^{3} + 3 \,{\left (88 \, B a b^{2} + 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (104 \, B a^{2} b + 59 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + 17 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{192 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(15*(8*B*a*b^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*(88*B*a*b^2 + 5*A*b^3)*x^3 + 2*(104*B*a^2*b + 59*A*a*b^2)
*x^2 + 8*(8*B*a^3 + 17*A*a^2*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(3/2)*x^4), 1/192*(
15*(8*B*a*b^3 - A*b^4)*x^4*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - (48*A*a^3 + 3*(8
8*B*a*b^2 + 5*A*b^3)*x^3 + 2*(104*B*a^2*b + 59*A*a*b^2)*x^2 + 8*(8*B*a^3 + 17*A*
a^2*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^4)]

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Sympy [A]  time = 159.828, size = 1481, normalized size = 10.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-558*A*a**6*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**5*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**4*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) - 198*A*a**4*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**3*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) + 240*A*a**3*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**3*b**4*sqrt(a**(-9))*log(-a**5
*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*A*a**3*b**4*sqrt(a**(-9))*log(a**5*sqrt
(a**(-9)) + sqrt(a + b*x))/128 - 90*A*a**2*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) - 30*A*a**2*b**4*sqrt(a
+ b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 15*A*a**2*b**4*sqrt(a**(
-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 15*A*a**2*b**4*sqrt(a**(-7))*
log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 18*A*a*b**4*(a + b*x)**(3/2)/(-8*a*
*4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 9*A*a*b**4*sqrt(a**(-5))*log(-a**3*sqr
t(a**(-5)) + sqrt(a + b*x))/8 - 9*A*a*b**4*sqrt(a**(-5))*log(a**3*sqrt(a**(-5))
+ sqrt(a + b*x))/8 - A*b**4*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x
))/2 + A*b**4*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - A*b**3*s
qrt(a + b*x)/(a*x) - 66*B*a**5*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**4*b**3*(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) - 30*B*a**3*
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) - 30*B*a**3*b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a
+ b*x)**2) - 5*B*a**3*b**3*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x)
)/16 + 5*B*a**3*b**3*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 +
18*B*a**2*b**3*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) +
9*B*a**2*b**3*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 9*B*a**
2*b**3*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*B*a*b**3*sqrt
(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + 3*B*a*b**3*sqrt(a**(-3))*
log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 2*B*b**3*Piecewise((-atan(sqrt(a + b
*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0)
& (a < a + b*x)), (atanh(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*
x))) - 3*B*b**2*sqrt(a + b*x)/x

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GIAC/XCAS [A]  time = 0.216125, size = 239, normalized size = 1.68 \[ \frac{\frac{15 \,{\left (8 \, B a b^{4} - A b^{5}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{264 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{4} - 584 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 440 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{4} - 120 \, \sqrt{b x + a} B a^{4} b^{4} + 15 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{5} + 73 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{5} - 55 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{5} + 15 \, \sqrt{b x + a} A a^{3} b^{5}}{a b^{4} x^{4}}}{192 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(15*(8*B*a*b^4 - A*b^5)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) - (264
*(b*x + a)^(7/2)*B*a*b^4 - 584*(b*x + a)^(5/2)*B*a^2*b^4 + 440*(b*x + a)^(3/2)*B
*a^3*b^4 - 120*sqrt(b*x + a)*B*a^4*b^4 + 15*(b*x + a)^(7/2)*A*b^5 + 73*(b*x + a)
^(5/2)*A*a*b^5 - 55*(b*x + a)^(3/2)*A*a^2*b^5 + 15*sqrt(b*x + a)*A*a^3*b^5)/(a*b
^4*x^4))/b