3.396 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=172 \[ \frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{7/2}}-\frac{3 b^3 \sqrt{a+b x} (A b-2 a B)}{128 a^3 x}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a^2 x^2}+\frac{(a+b x)^{3/2} (A b-2 a B)}{8 a x^4}+\frac{b \sqrt{a+b x} (A b-2 a B)}{16 a x^3}-\frac{A (a+b x)^{5/2}}{5 a x^5} \]

[Out]

(b*(A*b - 2*a*B)*Sqrt[a + b*x])/(16*a*x^3) + (b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(
64*a^2*x^2) - (3*b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(128*a^3*x) + ((A*b - 2*a*B)*(
a + b*x)^(3/2))/(8*a*x^4) - (A*(a + b*x)^(5/2))/(5*a*x^5) + (3*b^4*(A*b - 2*a*B)
*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(7/2))

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Rubi [A]  time = 0.233561, antiderivative size = 172, 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 \[ \frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{7/2}}-\frac{3 b^3 \sqrt{a+b x} (A b-2 a B)}{128 a^3 x}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a^2 x^2}+\frac{(a+b x)^{3/2} (A b-2 a B)}{8 a x^4}+\frac{b \sqrt{a+b x} (A b-2 a B)}{16 a x^3}-\frac{A (a+b x)^{5/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^6,x]

[Out]

(b*(A*b - 2*a*B)*Sqrt[a + b*x])/(16*a*x^3) + (b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(
64*a^2*x^2) - (3*b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(128*a^3*x) + ((A*b - 2*a*B)*(
a + b*x)^(3/2))/(8*a*x^4) - (A*(a + b*x)^(5/2))/(5*a*x^5) + (3*b^4*(A*b - 2*a*B)
*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(7/2))

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Rubi in Sympy [A]  time = 22.6135, size = 158, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{5 a x^{5}} + \frac{b \sqrt{a + b x} \left (A b - 2 B a\right )}{16 a x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (\frac{A b}{2} - B a\right )}{4 a x^{4}} + \frac{b^{2} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{32 a^{2} x^{2}} - \frac{3 b^{3} \sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{64 a^{3} x} + \frac{3 b^{4} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**6,x)

[Out]

-A*(a + b*x)**(5/2)/(5*a*x**5) + b*sqrt(a + b*x)*(A*b - 2*B*a)/(16*a*x**3) + (a
+ b*x)**(3/2)*(A*b/2 - B*a)/(4*a*x**4) + b**2*sqrt(a + b*x)*(A*b/2 - B*a)/(32*a*
*2*x**2) - 3*b**3*sqrt(a + b*x)*(A*b/2 - B*a)/(64*a**3*x) + 3*b**4*(A*b/2 - B*a)
*atanh(sqrt(a + b*x)/sqrt(a))/(64*a**(7/2))

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Mathematica [A]  time = 0.209933, size = 128, normalized size = 0.74 \[ \frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{7/2}}-\frac{\sqrt{a+b x} \left (32 a^4 (4 A+5 B x)+16 a^3 b x (11 A+15 B x)+4 a^2 b^2 x^2 (2 A+5 B x)-10 a b^3 x^3 (A+3 B x)+15 A b^4 x^4\right )}{640 a^3 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^6,x]

[Out]

-(Sqrt[a + b*x]*(15*A*b^4*x^4 - 10*a*b^3*x^3*(A + 3*B*x) + 4*a^2*b^2*x^2*(2*A +
5*B*x) + 32*a^4*(4*A + 5*B*x) + 16*a^3*b*x*(11*A + 15*B*x)))/(640*a^3*x^5) + (3*
b^4*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(7/2))

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Maple [A]  time = 0.021, size = 129, normalized size = 0.8 \[ 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ( -{\frac{ \left ( 3\,Ab-6\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{3}}}+{\frac{ \left ( 7\,Ab-14\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,{a}^{2}}}-1/10\,{\frac{Ab \left ( bx+a \right ) ^{5/2}}{a}}+ \left ( -{\frac{7\,Ab}{128}}+{\frac{7\,Ba}{64}} \right ) \left ( bx+a \right ) ^{3/2}+{\frac{3\,a \left ( Ab-2\,Ba \right ) \sqrt{bx+a}}{256}} \right ) }+{\frac{3\,Ab-6\,Ba}{256\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^6,x)

[Out]

2*b^4*((-3/256*(A*b-2*B*a)/a^3*(b*x+a)^(9/2)+7/128*(A*b-2*B*a)/a^2*(b*x+a)^(7/2)
-1/10*A*b/a*(b*x+a)^(5/2)+(-7/128*A*b+7/64*B*a)*(b*x+a)^(3/2)+3/256*a*(A*b-2*B*a
)*(b*x+a)^(1/2))/x^5/b^5+3/256*(A*b-2*B*a)/a^(7/2)*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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230262, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (2 \, B a b^{4} - A b^{5}\right )} x^{5} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (128 \, A a^{4} - 15 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{4} + 10 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{1280 \, a^{\frac{7}{2}} x^{5}}, \frac{15 \,{\left (2 \, B a b^{4} - A b^{5}\right )} x^{5} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (128 \, A a^{4} - 15 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{4} + 10 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{640 \, \sqrt{-a} a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[-1/1280*(15*(2*B*a*b^4 - A*b^5)*x^5*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*
a)/x) + 2*(128*A*a^4 - 15*(2*B*a*b^3 - A*b^4)*x^4 + 10*(2*B*a^2*b^2 - A*a*b^3)*x
^3 + 8*(30*B*a^3*b + A*a^2*b^2)*x^2 + 16*(10*B*a^4 + 11*A*a^3*b)*x)*sqrt(b*x + a
)*sqrt(a))/(a^(7/2)*x^5), 1/640*(15*(2*B*a*b^4 - A*b^5)*x^5*arctan(a/(sqrt(b*x +
a)*sqrt(-a))) - (128*A*a^4 - 15*(2*B*a*b^3 - A*b^4)*x^4 + 10*(2*B*a^2*b^2 - A*a
*b^3)*x^3 + 8*(30*B*a^3*b + A*a^2*b^2)*x^2 + 16*(10*B*a^4 + 11*A*a^3*b)*x)*sqrt(
b*x + a)*sqrt(-a))/(sqrt(-a)*a^3*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)**(3/2)*(B*x+A)/x**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215287, size = 259, normalized size = 1.51 \[ \frac{\frac{15 \,{\left (2 \, B a b^{5} - A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{30 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} - 140 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 140 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 30 \, \sqrt{b x + a} B a^{5} b^{5} - 15 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 70 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} - 128 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 70 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 15 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{3} b^{5} x^{5}}}{640 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^6,x, algorithm="giac")

[Out]

1/640*(15*(2*B*a*b^5 - A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (3
0*(b*x + a)^(9/2)*B*a*b^5 - 140*(b*x + a)^(7/2)*B*a^2*b^5 + 140*(b*x + a)^(3/2)*
B*a^4*b^5 - 30*sqrt(b*x + a)*B*a^5*b^5 - 15*(b*x + a)^(9/2)*A*b^6 + 70*(b*x + a)
^(7/2)*A*a*b^6 - 128*(b*x + a)^(5/2)*A*a^2*b^6 - 70*(b*x + a)^(3/2)*A*a^3*b^6 +
15*sqrt(b*x + a)*A*a^4*b^6)/(a^3*b^5*x^5))/b