3.408 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=177 \[ -\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^3 \sqrt{a+b x} (3 A b-10 a B)}{128 a^2 x}+\frac{b^2 \sqrt{a+b x} (3 A b-10 a B)}{64 a x^2}+\frac{(a+b x)^{5/2} (3 A b-10 a B)}{40 a x^4}+\frac{b (a+b x)^{3/2} (3 A b-10 a B)}{48 a x^3}-\frac{A (a+b x)^{7/2}}{5 a x^5} \]

[Out]

(b^2*(3*A*b - 10*a*B)*Sqrt[a + b*x])/(64*a*x^2) + (b^3*(3*A*b - 10*a*B)*Sqrt[a +
b*x])/(128*a^2*x) + (b*(3*A*b - 10*a*B)*(a + b*x)^(3/2))/(48*a*x^3) + ((3*A*b -
10*a*B)*(a + b*x)^(5/2))/(40*a*x^4) - (A*(a + b*x)^(7/2))/(5*a*x^5) - (b^4*(3*A
*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(5/2))

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Rubi [A]  time = 0.233506, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^3 \sqrt{a+b x} (3 A b-10 a B)}{128 a^2 x}+\frac{b^2 \sqrt{a+b x} (3 A b-10 a B)}{64 a x^2}+\frac{(a+b x)^{5/2} (3 A b-10 a B)}{40 a x^4}+\frac{b (a+b x)^{3/2} (3 A b-10 a B)}{48 a x^3}-\frac{A (a+b x)^{7/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^6,x]

[Out]

(b^2*(3*A*b - 10*a*B)*Sqrt[a + b*x])/(64*a*x^2) + (b^3*(3*A*b - 10*a*B)*Sqrt[a +
b*x])/(128*a^2*x) + (b*(3*A*b - 10*a*B)*(a + b*x)^(3/2))/(48*a*x^3) + ((3*A*b -
10*a*B)*(a + b*x)^(5/2))/(40*a*x^4) - (A*(a + b*x)^(7/2))/(5*a*x^5) - (b^4*(3*A
*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(5/2))

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Rubi in Sympy [A]  time = 20.869, size = 162, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{5 a x^{5}} + \frac{b^{2} \sqrt{a + b x} \left (3 A b - 10 B a\right )}{64 a x^{2}} + \frac{b \left (a + b x\right )^{\frac{3}{2}} \left (3 A b - 10 B a\right )}{48 a x^{3}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (3 A b - 10 B a\right )}{40 a x^{4}} + \frac{b^{3} \sqrt{a + b x} \left (3 A b - 10 B a\right )}{128 a^{2} x} - \frac{b^{4} \left (3 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{128 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**6,x)

[Out]

-A*(a + b*x)**(7/2)/(5*a*x**5) + b**2*sqrt(a + b*x)*(3*A*b - 10*B*a)/(64*a*x**2)
+ b*(a + b*x)**(3/2)*(3*A*b - 10*B*a)/(48*a*x**3) + (a + b*x)**(5/2)*(3*A*b - 1
0*B*a)/(40*a*x**4) + b**3*sqrt(a + b*x)*(3*A*b - 10*B*a)/(128*a**2*x) - b**4*(3*
A*b - 10*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(128*a**(5/2))

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Mathematica [A]  time = 0.226187, size = 129, normalized size = 0.73 \[ \frac{b^4 (10 a B-3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{5/2}}-\frac{\sqrt{a+b x} \left (96 a^4 (4 A+5 B x)+16 a^3 b x (63 A+85 B x)+4 a^2 b^2 x^2 (186 A+295 B x)+30 a b^3 x^3 (A+5 B x)-45 A b^4 x^4\right )}{1920 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^6,x]

[Out]

-(Sqrt[a + b*x]*(-45*A*b^4*x^4 + 30*a*b^3*x^3*(A + 5*B*x) + 96*a^4*(4*A + 5*B*x)
+ 16*a^3*b*x*(63*A + 85*B*x) + 4*a^2*b^2*x^2*(186*A + 295*B*x)))/(1920*a^2*x^5)
+ (b^4*(-3*A*b + 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(5/2))

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Maple [A]  time = 0.02, size = 140, normalized size = 0.8 \[ 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ({\frac{ \left ( 3\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{2}}}-{\frac{ \left ( 21\,Ab+58\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{384\,a}}+ \left ( -1/10\,Ab+1/3\,Ba \right ) \left ( bx+a \right ) ^{5/2}+{\frac{7\,a \left ( 3\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384}}-{\frac{{a}^{2} \left ( 3\,Ab-10\,Ba \right ) \sqrt{bx+a}}{256}} \right ) }-{\frac{3\,Ab-10\,Ba}{256\,{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^6,x)

[Out]

2*b^4*((1/256*(3*A*b-10*B*a)/a^2*(b*x+a)^(9/2)-1/384*(21*A*b+58*B*a)/a*(b*x+a)^(
7/2)+(-1/10*A*b+1/3*B*a)*(b*x+a)^(5/2)+7/384*a*(3*A*b-10*B*a)*(b*x+a)^(3/2)-1/25
6*a^2*(3*A*b-10*B*a)*(b*x+a)^(1/2))/x^5/b^5-1/256*(3*A*b-10*B*a)/a^(5/2)*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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219548, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} x^{5} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (384 \, A a^{4} + 15 \,{\left (10 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 10 \,{\left (118 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} + 8 \,{\left (170 \, B a^{3} b + 93 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} + 21 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{3840 \, a^{\frac{5}{2}} x^{5}}, -\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} x^{5} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (384 \, A a^{4} + 15 \,{\left (10 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 10 \,{\left (118 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{3} + 8 \,{\left (170 \, B a^{3} b + 93 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} + 21 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{1920 \, \sqrt{-a} a^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[-1/3840*(15*(10*B*a*b^4 - 3*A*b^5)*x^5*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x +
a)*a)/x) + 2*(384*A*a^4 + 15*(10*B*a*b^3 - 3*A*b^4)*x^4 + 10*(118*B*a^2*b^2 + 3*
A*a*b^3)*x^3 + 8*(170*B*a^3*b + 93*A*a^2*b^2)*x^2 + 48*(10*B*a^4 + 21*A*a^3*b)*x
)*sqrt(b*x + a)*sqrt(a))/(a^(5/2)*x^5), -1/1920*(15*(10*B*a*b^4 - 3*A*b^5)*x^5*a
rctan(a/(sqrt(b*x + a)*sqrt(-a))) + (384*A*a^4 + 15*(10*B*a*b^3 - 3*A*b^4)*x^4 +
10*(118*B*a^2*b^2 + 3*A*a*b^3)*x^3 + 8*(170*B*a^3*b + 93*A*a^2*b^2)*x^2 + 48*(1
0*B*a^4 + 21*A*a^3*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^5)]

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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**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220334, size = 281, normalized size = 1.59 \[ -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{150 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} + 580 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} - 1280 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{3} b^{5} + 700 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x + a} B a^{5} b^{5} - 45 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 210 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} + 384 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 210 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 45 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{2} b^{5} x^{5}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^6,x, algorithm="giac")

[Out]

-1/1920*(15*(10*B*a*b^5 - 3*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2)
+ (150*(b*x + a)^(9/2)*B*a*b^5 + 580*(b*x + a)^(7/2)*B*a^2*b^5 - 1280*(b*x + a)
^(5/2)*B*a^3*b^5 + 700*(b*x + a)^(3/2)*B*a^4*b^5 - 150*sqrt(b*x + a)*B*a^5*b^5 -
45*(b*x + a)^(9/2)*A*b^6 + 210*(b*x + a)^(7/2)*A*a*b^6 + 384*(b*x + a)^(5/2)*A*
a^2*b^6 - 210*(b*x + a)^(3/2)*A*a^3*b^6 + 45*sqrt(b*x + a)*A*a^4*b^6)/(a^2*b^5*x
^5))/b