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

Optimal. Leaf size=202 \[ -\frac{105 b^3 (11 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{13/2}}+\frac{105 b^2 \sqrt{a+b x} (11 A b-8 a B)}{64 a^6 x}-\frac{35 b \sqrt{a+b x} (11 A b-8 a B)}{32 a^5 x^2}+\frac{7 \sqrt{a+b x} (11 A b-8 a B)}{8 a^4 x^3}-\frac{3 (11 A b-8 a B)}{4 a^3 x^3 \sqrt{a+b x}}-\frac{11 A b-8 a B}{12 a^2 x^3 (a+b x)^{3/2}}-\frac{A}{4 a x^4 (a+b x)^{3/2}} \]

[Out]

-A/(4*a*x^4*(a + b*x)^(3/2)) - (11*A*b - 8*a*B)/(12*a^2*x^3*(a + b*x)^(3/2)) - (
3*(11*A*b - 8*a*B))/(4*a^3*x^3*Sqrt[a + b*x]) + (7*(11*A*b - 8*a*B)*Sqrt[a + b*x
])/(8*a^4*x^3) - (35*b*(11*A*b - 8*a*B)*Sqrt[a + b*x])/(32*a^5*x^2) + (105*b^2*(
11*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^6*x) - (105*b^3*(11*A*b - 8*a*B)*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]])/(64*a^(13/2))

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Rubi [A]  time = 0.28008, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{105 b^3 (11 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{13/2}}+\frac{105 b^2 \sqrt{a+b x} (11 A b-8 a B)}{64 a^6 x}-\frac{35 b \sqrt{a+b x} (11 A b-8 a B)}{32 a^5 x^2}+\frac{7 \sqrt{a+b x} (11 A b-8 a B)}{8 a^4 x^3}-\frac{3 (11 A b-8 a B)}{4 a^3 x^3 \sqrt{a+b x}}-\frac{11 A b-8 a B}{12 a^2 x^3 (a+b x)^{3/2}}-\frac{A}{4 a x^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(4*a*x^4*(a + b*x)^(3/2)) - (11*A*b - 8*a*B)/(12*a^2*x^3*(a + b*x)^(3/2)) - (
3*(11*A*b - 8*a*B))/(4*a^3*x^3*Sqrt[a + b*x]) + (7*(11*A*b - 8*a*B)*Sqrt[a + b*x
])/(8*a^4*x^3) - (35*b*(11*A*b - 8*a*B)*Sqrt[a + b*x])/(32*a^5*x^2) + (105*b^2*(
11*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^6*x) - (105*b^3*(11*A*b - 8*a*B)*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]])/(64*a^(13/2))

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Rubi in Sympy [A]  time = 27.9217, size = 197, normalized size = 0.98 \[ - \frac{A}{4 a x^{4} \left (a + b x\right )^{\frac{3}{2}}} - \frac{11 A b - 8 B a}{12 a^{2} x^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{3 \left (11 A b - 8 B a\right )}{4 a^{3} x^{3} \sqrt{a + b x}} + \frac{7 \sqrt{a + b x} \left (11 A b - 8 B a\right )}{8 a^{4} x^{3}} - \frac{35 b \sqrt{a + b x} \left (11 A b - 8 B a\right )}{32 a^{5} x^{2}} + \frac{105 b^{2} \sqrt{a + b x} \left (11 A b - 8 B a\right )}{64 a^{6} x} - \frac{105 b^{3} \left (11 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(4*a*x**4*(a + b*x)**(3/2)) - (11*A*b - 8*B*a)/(12*a**2*x**3*(a + b*x)**(3/2)
) - 3*(11*A*b - 8*B*a)/(4*a**3*x**3*sqrt(a + b*x)) + 7*sqrt(a + b*x)*(11*A*b - 8
*B*a)/(8*a**4*x**3) - 35*b*sqrt(a + b*x)*(11*A*b - 8*B*a)/(32*a**5*x**2) + 105*b
**2*sqrt(a + b*x)*(11*A*b - 8*B*a)/(64*a**6*x) - 105*b**3*(11*A*b - 8*B*a)*atanh
(sqrt(a + b*x)/sqrt(a))/(64*a**(13/2))

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Mathematica [A]  time = 0.414615, size = 151, normalized size = 0.75 \[ \frac{\frac{\sqrt{a} \left (-16 a^5 (3 A+4 B x)+8 a^4 b x (11 A+18 B x)-18 a^3 b^2 x^2 (11 A+28 B x)+21 a^2 b^3 x^3 (33 A-160 B x)+420 a b^4 x^4 (11 A-6 B x)+3465 A b^5 x^5\right )}{x^4 (a+b x)^{3/2}}+315 b^3 (8 a B-11 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{192 a^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(3465*A*b^5*x^5 + 21*a^2*b^3*x^3*(33*A - 160*B*x) + 420*a*b^4*x^4*(11*
A - 6*B*x) - 16*a^5*(3*A + 4*B*x) + 8*a^4*b*x*(11*A + 18*B*x) - 18*a^3*b^2*x^2*(
11*A + 28*B*x)))/(x^4*(a + b*x)^(3/2)) + 315*b^3*(-11*A*b + 8*a*B)*ArcTanh[Sqrt[
a + b*x]/Sqrt[a]])/(192*a^(13/2))

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Maple [A]  time = 0.029, size = 168, normalized size = 0.8 \[ 2\,{b}^{3} \left ( -{\frac{-5\,Ab+4\,Ba}{{a}^{6}\sqrt{bx+a}}}-1/3\,{\frac{-Ab+Ba}{{a}^{5} \left ( bx+a \right ) ^{3/2}}}+{\frac{1}{{a}^{6}} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ( \left ({\frac{515\,Ab}{128}}-{\frac{41\,Ba}{16}} \right ) \left ( bx+a \right ) ^{7/2}+ \left ( -{\frac{5153\,Aab}{384}}+{\frac{403\,B{a}^{2}}{48}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ({\frac{5855\,A{a}^{2}b}{384}}-{\frac{445\,B{a}^{3}}{48}} \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{765\,A{a}^{3}b}{128}}+{\frac{55\,B{a}^{4}}{16}} \right ) \sqrt{bx+a} \right ) }-{\frac{1155\,Ab-840\,Ba}{128\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^3*(-(-5*A*b+4*B*a)/a^6/(b*x+a)^(1/2)-1/3*(-A*b+B*a)/a^5/(b*x+a)^(3/2)+1/a^6*
(((515/128*A*b-41/16*B*a)*(b*x+a)^(7/2)+(-5153/384*A*a*b+403/48*B*a^2)*(b*x+a)^(
5/2)+(5855/384*A*a^2*b-445/48*B*a^3)*(b*x+a)^(3/2)+(-765/128*A*a^3*b+55/16*B*a^4
)*(b*x+a)^(1/2))/x^4/b^4-105/128*(11*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)^(5/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241924, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left ({\left (8 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} +{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4}\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{5} + 315 \,{\left (8 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 420 \,{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 63 \,{\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 18 \,{\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{a}}{384 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )} \sqrt{b x + a} \sqrt{a}}, -\frac{315 \,{\left ({\left (8 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} +{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4}\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (48 \, A a^{5} + 315 \,{\left (8 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 420 \,{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 63 \,{\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 18 \,{\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{-a}}{192 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

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*(315*((8*B*a*b^4 - 11*A*b^5)*x^5 + (8*B*a^2*b^3 - 11*A*a*b^4)*x^4)*sqrt(
b*x + a)*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*(48*A*a^5 + 315*(8
*B*a*b^4 - 11*A*b^5)*x^5 + 420*(8*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 63*(8*B*a^3*b^2
- 11*A*a^2*b^3)*x^3 - 18*(8*B*a^4*b - 11*A*a^3*b^2)*x^2 + 8*(8*B*a^5 - 11*A*a^4*
b)*x)*sqrt(a))/((a^6*b*x^5 + a^7*x^4)*sqrt(b*x + a)*sqrt(a)), -1/192*(315*((8*B*
a*b^4 - 11*A*b^5)*x^5 + (8*B*a^2*b^3 - 11*A*a*b^4)*x^4)*sqrt(b*x + a)*arctan(a/(
sqrt(b*x + a)*sqrt(-a))) + (48*A*a^5 + 315*(8*B*a*b^4 - 11*A*b^5)*x^5 + 420*(8*B
*a^2*b^3 - 11*A*a*b^4)*x^4 + 63*(8*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 - 18*(8*B*a^4*b
- 11*A*a^3*b^2)*x^2 + 8*(8*B*a^5 - 11*A*a^4*b)*x)*sqrt(-a))/((a^6*b*x^5 + a^7*x
^4)*sqrt(b*x + a)*sqrt(-a))]

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Sympy [A]  time = 121.471, size = 1137, normalized size = 5.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-48*a**(185/2)*b**201*x**201/(192*a**(189/2)*b**(403/2)*x**(411/2)*sqrt(a/(b*
x) + 1) + 192*a**(187/2)*b**(405/2)*x**(413/2)*sqrt(a/(b*x) + 1)) + 88*a**(183/2
)*b**202*x**202/(192*a**(189/2)*b**(403/2)*x**(411/2)*sqrt(a/(b*x) + 1) + 192*a*
*(187/2)*b**(405/2)*x**(413/2)*sqrt(a/(b*x) + 1)) - 198*a**(181/2)*b**203*x**203
/(192*a**(189/2)*b**(403/2)*x**(411/2)*sqrt(a/(b*x) + 1) + 192*a**(187/2)*b**(40
5/2)*x**(413/2)*sqrt(a/(b*x) + 1)) + 693*a**(179/2)*b**204*x**204/(192*a**(189/2
)*b**(403/2)*x**(411/2)*sqrt(a/(b*x) + 1) + 192*a**(187/2)*b**(405/2)*x**(413/2)
*sqrt(a/(b*x) + 1)) + 4620*a**(177/2)*b**205*x**205/(192*a**(189/2)*b**(403/2)*x
**(411/2)*sqrt(a/(b*x) + 1) + 192*a**(187/2)*b**(405/2)*x**(413/2)*sqrt(a/(b*x)
+ 1)) + 3465*a**(175/2)*b**206*x**206/(192*a**(189/2)*b**(403/2)*x**(411/2)*sqrt
(a/(b*x) + 1) + 192*a**(187/2)*b**(405/2)*x**(413/2)*sqrt(a/(b*x) + 1)) - 3465*a
**88*b**(411/2)*x**(411/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(1
92*a**(189/2)*b**(403/2)*x**(411/2)*sqrt(a/(b*x) + 1) + 192*a**(187/2)*b**(405/2
)*x**(413/2)*sqrt(a/(b*x) + 1)) - 3465*a**87*b**(413/2)*x**(413/2)*sqrt(a/(b*x)
+ 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(192*a**(189/2)*b**(403/2)*x**(411/2)*sqrt
(a/(b*x) + 1) + 192*a**(187/2)*b**(405/2)*x**(413/2)*sqrt(a/(b*x) + 1))) + B*(-8
*a**(133/2)*b**128*x**128/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1)
+ 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) + 18*a**(131/2)*b**129
*x**129/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b
**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) - 63*a**(129/2)*b**130*x**130/(24*a**(13
7/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/
2)*sqrt(a/(b*x) + 1)) - 420*a**(127/2)*b**131*x**131/(24*a**(137/2)*b**(257/2)*x
**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) +
1)) - 315*a**(125/2)*b**132*x**132/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/
(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) + 315*a**63*
b**(263/2)*x**(263/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a**
(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(2
65/2)*sqrt(a/(b*x) + 1)) + 315*a**62*b**(265/2)*x**(265/2)*sqrt(a/(b*x) + 1)*asi
nh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x)
+ 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)))

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GIAC/XCAS [A]  time = 0.220759, size = 301, normalized size = 1.49 \[ -\frac{105 \,{\left (8 \, B a b^{3} - 11 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a^{6}} - \frac{2 \,{\left (12 \,{\left (b x + a\right )} B a b^{3} + B a^{2} b^{3} - 15 \,{\left (b x + a\right )} A b^{4} - A a b^{4}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}} - \frac{984 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{3} - 3224 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{3} + 3560 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{3} - 1320 \, \sqrt{b x + a} B a^{4} b^{3} - 1545 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{4} + 5153 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{4} - 5855 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{4} + 2295 \, \sqrt{b x + a} A a^{3} b^{4}}{192 \, a^{6} b^{4} x^{4}} \]

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]

-105/64*(8*B*a*b^3 - 11*A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^6) - 2
/3*(12*(b*x + a)*B*a*b^3 + B*a^2*b^3 - 15*(b*x + a)*A*b^4 - A*a*b^4)/((b*x + a)^
(3/2)*a^6) - 1/192*(984*(b*x + a)^(7/2)*B*a*b^3 - 3224*(b*x + a)^(5/2)*B*a^2*b^3
+ 3560*(b*x + a)^(3/2)*B*a^3*b^3 - 1320*sqrt(b*x + a)*B*a^4*b^3 - 1545*(b*x + a
)^(7/2)*A*b^4 + 5153*(b*x + a)^(5/2)*A*a*b^4 - 5855*(b*x + a)^(3/2)*A*a^2*b^4 +
2295*sqrt(b*x + a)*A*a^3*b^4)/(a^6*b^4*x^4)