3.420 \(\int \frac{A+B x}{x^5 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}+\frac{5 b^2 \sqrt{a+b x} (7 A b-8 a B)}{64 a^4 x}-\frac{5 b \sqrt{a+b x} (7 A b-8 a B)}{96 a^3 x^2}+\frac{\sqrt{a+b x} (7 A b-8 a B)}{24 a^2 x^3}-\frac{A \sqrt{a+b x}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.191999, antiderivative size = 146, 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 \[ -\frac{5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}+\frac{5 b^2 \sqrt{a+b x} (7 A b-8 a B)}{64 a^4 x}-\frac{5 b \sqrt{a+b x} (7 A b-8 a B)}{96 a^3 x^2}+\frac{\sqrt{a+b x} (7 A b-8 a B)}{24 a^2 x^3}-\frac{A \sqrt{a+b x}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.2389, size = 141, normalized size = 0.97 \[ - \frac{A \sqrt{a + b x}}{4 a x^{4}} + \frac{\sqrt{a + b x} \left (7 A b - 8 B a\right )}{24 a^{2} x^{3}} - \frac{5 b \sqrt{a + b x} \left (7 A b - 8 B a\right )}{96 a^{3} x^{2}} + \frac{5 b^{2} \sqrt{a + b x} \left (7 A b - 8 B a\right )}{64 a^{4} x} - \frac{5 b^{3} \left (7 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**5/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.213222, size = 112, normalized size = 0.77 \[ \frac{5 b^3 (8 a B-7 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{9/2}}+\frac{\sqrt{a+b x} \left (-16 a^3 (3 A+4 B x)+8 a^2 b x (7 A+10 B x)-10 a b^2 x^2 (7 A+12 B x)+105 A b^3 x^3\right )}{192 a^4 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(105*A*b^3*x^3 - 16*a^3*(3*A + 4*B*x) + 8*a^2*b*x*(7*A + 10*B*x)
- 10*a*b^2*x^2*(7*A + 12*B*x)))/(192*a^4*x^4) + (5*b^3*(-7*A*b + 8*a*B)*ArcTanh[
Sqrt[a + b*x]/Sqrt[a]])/(64*a^(9/2))

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Maple [A]  time = 0.019, size = 125, normalized size = 0.9 \[ 2\,{b}^{3} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ({\frac{ \left ( 35\,Ab-40\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,{a}^{4}}}-{\frac{ \left ( 385\,Ab-440\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{384\,{a}^{3}}}+{\frac{ \left ( 511\,Ab-584\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,{a}^{2}}}-{\frac{ \left ( 93\,Ab-88\,Ba \right ) \sqrt{bx+a}}{128\,a}} \right ) }-{\frac{35\,Ab-40\,Ba}{128\,{a}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^5/(b*x+a)^(1/2),x)

[Out]

2*b^3*((5/128*(7*A*b-8*B*a)/a^4*(b*x+a)^(7/2)-55/384/a^3*(7*A*b-8*B*a)*(b*x+a)^(
5/2)+73/384/a^2*(7*A*b-8*B*a)*(b*x+a)^(3/2)-1/128*(93*A*b-88*B*a)/a*(b*x+a)^(1/2
))/x^4/b^4-5/128*(7*A*b-8*B*a)/a^(9/2)*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)/(sqrt(b*x + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23173, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{3} + 15 \,{\left (8 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \,{\left (8 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{384 \, a^{\frac{9}{2}} x^{4}}, -\frac{15 \,{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (48 \, A a^{3} + 15 \,{\left (8 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \,{\left (8 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{192 \, \sqrt{-a} a^{4} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(15*(8*B*a*b^3 - 7*A*b^4)*x^4*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)
*a)/x) + 2*(48*A*a^3 + 15*(8*B*a*b^2 - 7*A*b^3)*x^3 - 10*(8*B*a^2*b - 7*A*a*b^2)
*x^2 + 8*(8*B*a^3 - 7*A*a^2*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(9/2)*x^4), -1/192*(
15*(8*B*a*b^3 - 7*A*b^4)*x^4*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (48*A*a^3 + 15
*(8*B*a*b^2 - 7*A*b^3)*x^3 - 10*(8*B*a^2*b - 7*A*a*b^2)*x^2 + 8*(8*B*a^3 - 7*A*a
^2*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^4*x^4)]

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Sympy [A]  time = 146.252, size = 303, normalized size = 2.08 \[ - \frac{A}{4 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{24 a x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{7 A b^{\frac{3}{2}}}{96 a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 A b^{\frac{5}{2}}}{192 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{35 A b^{\frac{7}{2}}}{64 a^{4} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{35 A b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64 a^{\frac{9}{2}}} - \frac{B}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{B \sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 B b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 B b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 B b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**5/(b*x+a)**(1/2),x)

[Out]

-A/(4*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) + A*sqrt(b)/(24*a*x**(7/2)*sqrt(a/(b*x
) + 1)) - 7*A*b**(3/2)/(96*a**2*x**(5/2)*sqrt(a/(b*x) + 1)) + 35*A*b**(5/2)/(192
*a**3*x**(3/2)*sqrt(a/(b*x) + 1)) + 35*A*b**(7/2)/(64*a**4*sqrt(x)*sqrt(a/(b*x)
+ 1)) - 35*A*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a**(9/2)) - B/(3*sqrt(b)*
x**(7/2)*sqrt(a/(b*x) + 1)) + B*sqrt(b)/(12*a*x**(5/2)*sqrt(a/(b*x) + 1)) - 5*B*
b**(3/2)/(24*a**2*x**(3/2)*sqrt(a/(b*x) + 1)) - 5*B*b**(5/2)/(8*a**3*sqrt(x)*sqr
t(a/(b*x) + 1)) + 5*B*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**(7/2))

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GIAC/XCAS [A]  time = 0.212373, size = 238, normalized size = 1.63 \[ -\frac{\frac{15 \,{\left (8 \, B a b^{4} - 7 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{120 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{4} - 440 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 584 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{4} - 264 \, \sqrt{b x + a} B a^{4} b^{4} - 105 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{5} + 385 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{5} - 511 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{5} + 279 \, \sqrt{b x + a} A a^{3} b^{5}}{a^{4} b^{4} x^{4}}}{192 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(15*(8*B*a*b^4 - 7*A*b^5)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) +
(120*(b*x + a)^(7/2)*B*a*b^4 - 440*(b*x + a)^(5/2)*B*a^2*b^4 + 584*(b*x + a)^(3
/2)*B*a^3*b^4 - 264*sqrt(b*x + a)*B*a^4*b^4 - 105*(b*x + a)^(7/2)*A*b^5 + 385*(b
*x + a)^(5/2)*A*a*b^5 - 511*(b*x + a)^(3/2)*A*a^2*b^5 + 279*sqrt(b*x + a)*A*a^3*
b^5)/(a^4*b^4*x^4))/b