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

Optimal. Leaf size=174 \[ -\frac{35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}+\frac{35 b^2 \sqrt{a+b x} (9 A b-8 a B)}{64 a^5 x}-\frac{35 b \sqrt{a+b x} (9 A b-8 a B)}{96 a^4 x^2}+\frac{7 \sqrt{a+b x} (9 A b-8 a B)}{24 a^3 x^3}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}-\frac{A}{4 a x^4 \sqrt{a+b x}} \]

[Out]

-A/(4*a*x^4*Sqrt[a + b*x]) - (9*A*b - 8*a*B)/(4*a^2*x^3*Sqrt[a + b*x]) + (7*(9*A
*b - 8*a*B)*Sqrt[a + b*x])/(24*a^3*x^3) - (35*b*(9*A*b - 8*a*B)*Sqrt[a + b*x])/(
96*a^4*x^2) + (35*b^2*(9*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^5*x) - (35*b^3*(9*A*b
- 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(11/2))

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Rubi [A]  time = 0.240391, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}+\frac{35 b^2 \sqrt{a+b x} (9 A b-8 a B)}{64 a^5 x}-\frac{35 b \sqrt{a+b x} (9 A b-8 a B)}{96 a^4 x^2}+\frac{7 \sqrt{a+b x} (9 A b-8 a B)}{24 a^3 x^3}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}-\frac{A}{4 a x^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(4*a*x^4*Sqrt[a + b*x]) - (9*A*b - 8*a*B)/(4*a^2*x^3*Sqrt[a + b*x]) + (7*(9*A
*b - 8*a*B)*Sqrt[a + b*x])/(24*a^3*x^3) - (35*b*(9*A*b - 8*a*B)*Sqrt[a + b*x])/(
96*a^4*x^2) + (35*b^2*(9*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^5*x) - (35*b^3*(9*A*b
- 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(11/2))

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Rubi in Sympy [A]  time = 22.3211, size = 170, normalized size = 0.98 \[ - \frac{A}{4 a x^{4} \sqrt{a + b x}} - \frac{9 A b - 8 B a}{4 a^{2} x^{3} \sqrt{a + b x}} + \frac{7 \sqrt{a + b x} \left (9 A b - 8 B a\right )}{24 a^{3} x^{3}} - \frac{35 b \sqrt{a + b x} \left (9 A b - 8 B a\right )}{96 a^{4} x^{2}} + \frac{35 b^{2} \sqrt{a + b x} \left (9 A b - 8 B a\right )}{64 a^{5} x} - \frac{35 b^{3} \left (9 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(4*a*x**4*sqrt(a + b*x)) - (9*A*b - 8*B*a)/(4*a**2*x**3*sqrt(a + b*x)) + 7*sq
rt(a + b*x)*(9*A*b - 8*B*a)/(24*a**3*x**3) - 35*b*sqrt(a + b*x)*(9*A*b - 8*B*a)/
(96*a**4*x**2) + 35*b**2*sqrt(a + b*x)*(9*A*b - 8*B*a)/(64*a**5*x) - 35*b**3*(9*
A*b - 8*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(64*a**(11/2))

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Mathematica [A]  time = 0.294283, size = 131, normalized size = 0.75 \[ \frac{35 b^3 (8 a B-9 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}+\frac{-16 a^4 (3 A+4 B x)+8 a^3 b x (9 A+14 B x)-14 a^2 b^2 x^2 (9 A+20 B x)+105 a b^3 x^3 (3 A-8 B x)+945 A b^4 x^4}{192 a^5 x^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(945*A*b^4*x^4 + 105*a*b^3*x^3*(3*A - 8*B*x) - 16*a^4*(3*A + 4*B*x) + 8*a^3*b*x*
(9*A + 14*B*x) - 14*a^2*b^2*x^2*(9*A + 20*B*x))/(192*a^5*x^4*Sqrt[a + b*x]) + (3
5*b^3*(-9*A*b + 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(11/2))

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Maple [A]  time = 0.026, size = 147, normalized size = 0.8 \[ 2\,{b}^{3} \left ( -{\frac{-Ab+Ba}{{a}^{5}\sqrt{bx+a}}}+{\frac{1}{{a}^{5}} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ( \left ({\frac{187\,Ab}{128}}-{\frac{19\,Ba}{16}} \right ) \left ( bx+a \right ) ^{7/2}+ \left ( -{\frac{643\,Aab}{128}}+{\frac{193\,B{a}^{2}}{48}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ({\frac{765\,A{a}^{2}b}{128}}-{\frac{223\,B{a}^{3}}{48}} \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{325\,A{a}^{3}b}{128}}+{\frac{29\,B{a}^{4}}{16}} \right ) \sqrt{bx+a} \right ) }-{\frac{315\,Ab-280\,Ba}{128\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^3*(-1/a^5*(-A*b+B*a)/(b*x+a)^(1/2)+1/a^5*(((187/128*A*b-19/16*B*a)*(b*x+a)^(
7/2)+(-643/128*A*a*b+193/48*B*a^2)*(b*x+a)^(5/2)+(765/128*A*a^2*b-223/48*B*a^3)*
(b*x+a)^(3/2)+(-325/128*A*a^3*b+29/16*B*a^4)*(b*x+a)^(1/2))/x^4/b^4-35/128*(9*A*
b-8*B*a)/a^(1/2)*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 = 0.233475, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} \sqrt{b x + a} x^{4} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{4} + 105 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 35 \,{\left (8 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \,{\left (8 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{a}}{384 \, \sqrt{b x + a} a^{\frac{11}{2}} x^{4}}, -\frac{105 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} \sqrt{b x + a} x^{4} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (48 \, A a^{4} + 105 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 35 \,{\left (8 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \,{\left (8 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{-a}}{192 \, \sqrt{b x + a} \sqrt{-a} a^{5} x^{4}}\right ] \]

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*(105*(8*B*a*b^3 - 9*A*b^4)*sqrt(b*x + a)*x^4*log(((b*x + 2*a)*sqrt(a) -
2*sqrt(b*x + a)*a)/x) + 2*(48*A*a^4 + 105*(8*B*a*b^3 - 9*A*b^4)*x^4 + 35*(8*B*a^
2*b^2 - 9*A*a*b^3)*x^3 - 14*(8*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(8*B*a^4 - 9*A*a^3
*b)*x)*sqrt(a))/(sqrt(b*x + a)*a^(11/2)*x^4), -1/192*(105*(8*B*a*b^3 - 9*A*b^4)*
sqrt(b*x + a)*x^4*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (48*A*a^4 + 105*(8*B*a*b^
3 - 9*A*b^4)*x^4 + 35*(8*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*(8*B*a^3*b - 9*A*a^2*b^
2)*x^2 + 8*(8*B*a^4 - 9*A*a^3*b)*x)*sqrt(-a))/(sqrt(b*x + a)*sqrt(-a)*a^5*x^4)]

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Sympy [A]  time = 81.554, size = 301, normalized size = 1.73 \[ A \left (- \frac{1}{4 a \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 \sqrt{b}}{8 a^{2} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{21 b^{\frac{3}{2}}}{32 a^{3} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{105 b^{\frac{5}{2}}}{64 a^{4} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{315 b^{\frac{7}{2}}}{64 a^{5} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{315 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64 a^{\frac{11}{2}}}\right ) + B \left (- \frac{1}{3 a \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{7 \sqrt{b}}{12 a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{35 b^{\frac{3}{2}}}{24 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{35 b^{\frac{5}{2}}}{8 a^{4} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{9}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/(4*a*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) + 3*sqrt(b)/(8*a**2*x**(7/2)*sqrt
(a/(b*x) + 1)) - 21*b**(3/2)/(32*a**3*x**(5/2)*sqrt(a/(b*x) + 1)) + 105*b**(5/2)
/(64*a**4*x**(3/2)*sqrt(a/(b*x) + 1)) + 315*b**(7/2)/(64*a**5*sqrt(x)*sqrt(a/(b*
x) + 1)) - 315*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a**(11/2))) + B*(-1/(3*
a*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) + 7*sqrt(b)/(12*a**2*x**(5/2)*sqrt(a/(b*x)
+ 1)) - 35*b**(3/2)/(24*a**3*x**(3/2)*sqrt(a/(b*x) + 1)) - 35*b**(5/2)/(8*a**4*
sqrt(x)*sqrt(a/(b*x) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**(9/2
)))

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GIAC/XCAS [A]  time = 0.215155, size = 266, normalized size = 1.53 \[ -\frac{35 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a^{5}} - \frac{2 \,{\left (B a b^{3} - A b^{4}\right )}}{\sqrt{b x + a} a^{5}} - \frac{456 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{3} - 1544 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{3} + 1784 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{3} - 696 \, \sqrt{b x + a} B a^{4} b^{3} - 561 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{4} + 1929 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{4} - 2295 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{4} + 975 \, \sqrt{b x + a} A a^{3} b^{4}}{192 \, a^{5} b^{4} x^{4}} \]

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]

-35/64*(8*B*a*b^3 - 9*A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^5) - 2*(
B*a*b^3 - A*b^4)/(sqrt(b*x + a)*a^5) - 1/192*(456*(b*x + a)^(7/2)*B*a*b^3 - 1544
*(b*x + a)^(5/2)*B*a^2*b^3 + 1784*(b*x + a)^(3/2)*B*a^3*b^3 - 696*sqrt(b*x + a)*
B*a^4*b^3 - 561*(b*x + a)^(7/2)*A*b^4 + 1929*(b*x + a)^(5/2)*A*a*b^4 - 2295*(b*x
+ a)^(3/2)*A*a^2*b^4 + 975*sqrt(b*x + a)*A*a^3*b^4)/(a^5*b^4*x^4)