3.410 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=232 \[ -\frac{5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}+\frac{5 b^5 \sqrt{a+b x} (A b-2 a B)}{1024 a^4 x}-\frac{5 b^4 \sqrt{a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac{b^3 \sqrt{a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a x^4}+\frac{(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}+\frac{b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}-\frac{A (a+b x)^{7/2}}{7 a x^7} \]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])
/(384*a^2*x^3) - (5*b^4*(A*b - 2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*
b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a + b*x)^(3/2))/(24*a
*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^
7) - (5*b^6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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Rubi [A]  time = 0.325765, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}+\frac{5 b^5 \sqrt{a+b x} (A b-2 a B)}{1024 a^4 x}-\frac{5 b^4 \sqrt{a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac{b^3 \sqrt{a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a x^4}+\frac{(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}+\frac{b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}-\frac{A (a+b x)^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])
/(384*a^2*x^3) - (5*b^4*(A*b - 2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*
b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a + b*x)^(3/2))/(24*a
*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^
7) - (5*b^6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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Rubi in Sympy [A]  time = 32.7061, size = 216, normalized size = 0.93 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{7 a x^{7}} + \frac{b^{2} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{32 a x^{4}} + \frac{b \left (a + b x\right )^{\frac{3}{2}} \left (A b - 2 B a\right )}{24 a x^{5}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (\frac{A b}{2} - B a\right )}{6 a x^{6}} + \frac{b^{3} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{192 a^{2} x^{3}} - \frac{5 b^{4} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{768 a^{3} x^{2}} + \frac{5 b^{5} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{512 a^{4} x} - \frac{5 b^{6} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{512 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**8,x)

[Out]

-A*(a + b*x)**(7/2)/(7*a*x**7) + b**2*sqrt(a + b*x)*(A*b/2 - B*a)/(32*a*x**4) +
b*(a + b*x)**(3/2)*(A*b - 2*B*a)/(24*a*x**5) + (a + b*x)**(5/2)*(A*b/2 - B*a)/(6
*a*x**6) + b**3*sqrt(a + b*x)*(A*b/2 - B*a)/(192*a**2*x**3) - 5*b**4*sqrt(a + b*
x)*(A*b/2 - B*a)/(768*a**3*x**2) + 5*b**5*sqrt(a + b*x)*(A*b/2 - B*a)/(512*a**4*
x) - 5*b**6*(A*b/2 - B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(512*a**(9/2))

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Mathematica [A]  time = 0.282374, size = 167, normalized size = 0.72 \[ \frac{5 b^6 (2 a B-A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}-\frac{\sqrt{a+b x} \left (512 a^6 (6 A+7 B x)+256 a^5 b x (29 A+35 B x)+32 a^4 b^2 x^2 (148 A+189 B x)+16 a^3 b^3 x^3 (3 A+7 B x)-28 a^2 b^4 x^4 (2 A+5 B x)+70 a b^5 x^5 (A+3 B x)-105 A b^6 x^6\right )}{21504 a^4 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

-(Sqrt[a + b*x]*(-105*A*b^6*x^6 + 70*a*b^5*x^5*(A + 3*B*x) - 28*a^2*b^4*x^4*(2*A
+ 5*B*x) + 16*a^3*b^3*x^3*(3*A + 7*B*x) + 512*a^6*(6*A + 7*B*x) + 256*a^5*b*x*(
29*A + 35*B*x) + 32*a^4*b^2*x^2*(148*A + 189*B*x)))/(21504*a^4*x^7) + (5*b^6*(-(
A*b) + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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Maple [A]  time = 0.023, size = 169, normalized size = 0.7 \[ 2\,{b}^{6} \left ({\frac{1}{{b}^{7}{x}^{7}} \left ({\frac{ \left ( 5\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{13/2}}{2048\,{a}^{4}}}-{\frac{ \left ( 25\,Ab-50\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1536\,{a}^{3}}}+{\frac{ \left ( 283\,Ab-566\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{6144\,{a}^{2}}}-1/14\,{\frac{Ab \left ( bx+a \right ) ^{7/2}}{a}}+ \left ( -{\frac{283\,Ab}{6144}}+{\frac{283\,Ba}{3072}} \right ) \left ( bx+a \right ) ^{5/2}+{\frac{25\,a \left ( Ab-2\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{1536}}-{\frac{5\,{a}^{2} \left ( Ab-2\,Ba \right ) \sqrt{bx+a}}{2048}} \right ) }-{\frac{5\,Ab-10\,Ba}{2048\,{a}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^8,x)

[Out]

2*b^6*((5/2048*(A*b-2*B*a)/a^4*(b*x+a)^(13/2)-25/1536*(A*b-2*B*a)/a^3*(b*x+a)^(1
1/2)+283/6144*(A*b-2*B*a)/a^2*(b*x+a)^(9/2)-1/14*A*b/a*(b*x+a)^(7/2)+(-283/6144*
A*b+283/3072*B*a)*(b*x+a)^(5/2)+25/1536*a*(A*b-2*B*a)*(b*x+a)^(3/2)-5/2048*a^2*(
A*b-2*B*a)*(b*x+a)^(1/2))/x^7/b^7-5/2048*(A*b-2*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)*(b*x + a)^(5/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223688, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (2 \, B a b^{6} - A b^{7}\right )} x^{7} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3072 \, A a^{6} + 105 \,{\left (2 \, B a b^{5} - A b^{6}\right )} x^{6} - 70 \,{\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} x^{5} + 56 \,{\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{4} + 48 \,{\left (126 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{3} + 128 \,{\left (70 \, B a^{5} b + 37 \, A a^{4} b^{2}\right )} x^{2} + 256 \,{\left (14 \, B a^{6} + 29 \, A a^{5} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{43008 \, a^{\frac{9}{2}} x^{7}}, -\frac{105 \,{\left (2 \, B a b^{6} - A b^{7}\right )} x^{7} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3072 \, A a^{6} + 105 \,{\left (2 \, B a b^{5} - A b^{6}\right )} x^{6} - 70 \,{\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} x^{5} + 56 \,{\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{4} + 48 \,{\left (126 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{3} + 128 \,{\left (70 \, B a^{5} b + 37 \, A a^{4} b^{2}\right )} x^{2} + 256 \,{\left (14 \, B a^{6} + 29 \, A a^{5} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{21504 \, \sqrt{-a} a^{4} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^8,x, algorithm="fricas")

[Out]

[-1/43008*(105*(2*B*a*b^6 - A*b^7)*x^7*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a
)*a)/x) + 2*(3072*A*a^6 + 105*(2*B*a*b^5 - A*b^6)*x^6 - 70*(2*B*a^2*b^4 - A*a*b^
5)*x^5 + 56*(2*B*a^3*b^3 - A*a^2*b^4)*x^4 + 48*(126*B*a^4*b^2 + A*a^3*b^3)*x^3 +
128*(70*B*a^5*b + 37*A*a^4*b^2)*x^2 + 256*(14*B*a^6 + 29*A*a^5*b)*x)*sqrt(b*x +
a)*sqrt(a))/(a^(9/2)*x^7), -1/21504*(105*(2*B*a*b^6 - A*b^7)*x^7*arctan(a/(sqrt
(b*x + a)*sqrt(-a))) + (3072*A*a^6 + 105*(2*B*a*b^5 - A*b^6)*x^6 - 70*(2*B*a^2*b
^4 - A*a*b^5)*x^5 + 56*(2*B*a^3*b^3 - A*a^2*b^4)*x^4 + 48*(126*B*a^4*b^2 + A*a^3
*b^3)*x^3 + 128*(70*B*a^5*b + 37*A*a^4*b^2)*x^2 + 256*(14*B*a^6 + 29*A*a^5*b)*x)
*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^4*x^7)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(B*x+A)/x**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237674, size = 346, normalized size = 1.49 \[ -\frac{\frac{105 \,{\left (2 \, B a b^{7} - A b^{8}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{210 \,{\left (b x + a\right )}^{\frac{13}{2}} B a b^{7} - 1400 \,{\left (b x + a\right )}^{\frac{11}{2}} B a^{2} b^{7} + 3962 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{3} b^{7} - 3962 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{5} b^{7} + 1400 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{6} b^{7} - 210 \, \sqrt{b x + a} B a^{7} b^{7} - 105 \,{\left (b x + a\right )}^{\frac{13}{2}} A b^{8} + 700 \,{\left (b x + a\right )}^{\frac{11}{2}} A a b^{8} - 1981 \,{\left (b x + a\right )}^{\frac{9}{2}} A a^{2} b^{8} + 3072 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{3} b^{8} + 1981 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{4} b^{8} - 700 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{5} b^{8} + 105 \, \sqrt{b x + a} A a^{6} b^{8}}{a^{4} b^{7} x^{7}}}{21504 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^8,x, algorithm="giac")

[Out]

-1/21504*(105*(2*B*a*b^7 - A*b^8)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4)
+ (210*(b*x + a)^(13/2)*B*a*b^7 - 1400*(b*x + a)^(11/2)*B*a^2*b^7 + 3962*(b*x +
a)^(9/2)*B*a^3*b^7 - 3962*(b*x + a)^(5/2)*B*a^5*b^7 + 1400*(b*x + a)^(3/2)*B*a^6
*b^7 - 210*sqrt(b*x + a)*B*a^7*b^7 - 105*(b*x + a)^(13/2)*A*b^8 + 700*(b*x + a)^
(11/2)*A*a*b^8 - 1981*(b*x + a)^(9/2)*A*a^2*b^8 + 3072*(b*x + a)^(7/2)*A*a^3*b^8
+ 1981*(b*x + a)^(5/2)*A*a^4*b^8 - 700*(b*x + a)^(3/2)*A*a^5*b^8 + 105*sqrt(b*x
+ a)*A*a^6*b^8)/(a^4*b^7*x^7))/b