3.552 \(\int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx\)

Optimal. Leaf size=152 \[ \frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 b^2 \sqrt{a+b x^2} (A b-8 a B)}{128 a x^2}+\frac{\left (a+b x^2\right )^{5/2} (A b-8 a B)}{48 a x^6}+\frac{5 b \left (a+b x^2\right )^{3/2} (A b-8 a B)}{192 a x^4}-\frac{A \left (a+b x^2\right )^{7/2}}{8 a x^8} \]

[Out]

(5*b^2*(A*b - 8*a*B)*Sqrt[a + b*x^2])/(128*a*x^2) + (5*b*(A*b - 8*a*B)*(a + b*x^
2)^(3/2))/(192*a*x^4) + ((A*b - 8*a*B)*(a + b*x^2)^(5/2))/(48*a*x^6) - (A*(a + b
*x^2)^(7/2))/(8*a*x^8) + (5*b^3*(A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/
(128*a^(3/2))

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Rubi [A]  time = 0.302208, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 b^2 \sqrt{a+b x^2} (A b-8 a B)}{128 a x^2}+\frac{\left (a+b x^2\right )^{5/2} (A b-8 a B)}{48 a x^6}+\frac{5 b \left (a+b x^2\right )^{3/2} (A b-8 a B)}{192 a x^4}-\frac{A \left (a+b x^2\right )^{7/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^9,x]

[Out]

(5*b^2*(A*b - 8*a*B)*Sqrt[a + b*x^2])/(128*a*x^2) + (5*b*(A*b - 8*a*B)*(a + b*x^
2)^(3/2))/(192*a*x^4) + ((A*b - 8*a*B)*(a + b*x^2)^(5/2))/(48*a*x^6) - (A*(a + b
*x^2)^(7/2))/(8*a*x^8) + (5*b^3*(A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/
(128*a^(3/2))

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Rubi in Sympy [A]  time = 25.2564, size = 139, normalized size = 0.91 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{8 a x^{8}} + \frac{5 b^{2} \sqrt{a + b x^{2}} \left (A b - 8 B a\right )}{128 a x^{2}} + \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - 8 B a\right )}{192 a x^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b - 8 B a\right )}{48 a x^{6}} + \frac{5 b^{3} \left (A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**9,x)

[Out]

-A*(a + b*x**2)**(7/2)/(8*a*x**8) + 5*b**2*sqrt(a + b*x**2)*(A*b - 8*B*a)/(128*a
*x**2) + 5*b*(a + b*x**2)**(3/2)*(A*b - 8*B*a)/(192*a*x**4) + (a + b*x**2)**(5/2
)*(A*b - 8*B*a)/(48*a*x**6) + 5*b**3*(A*b - 8*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a
))/(128*a**(3/2))

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Mathematica [A]  time = 0.272862, size = 143, normalized size = 0.94 \[ \frac{5 b^3 (A b-8 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{128 a^{3/2}}-\frac{5 b^3 \log (x) (A b-8 a B)}{128 a^{3/2}}+\sqrt{a+b x^2} \left (-\frac{a^2 A}{8 x^8}-\frac{b^2 (88 a B+5 A b)}{128 a x^2}-\frac{a (8 a B+17 A b)}{48 x^6}-\frac{b (104 a B+59 A b)}{192 x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^9,x]

[Out]

(-(a^2*A)/(8*x^8) - (a*(17*A*b + 8*a*B))/(48*x^6) - (b*(59*A*b + 104*a*B))/(192*
x^4) - (b^2*(5*A*b + 88*a*B))/(128*a*x^2))*Sqrt[a + b*x^2] - (5*b^3*(A*b - 8*a*B
)*Log[x])/(128*a^(3/2)) + (5*b^3*(A*b - 8*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])
/(128*a^(3/2))

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Maple [B]  time = 0.015, size = 311, normalized size = 2.1 \[ -{\frac{A}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ab}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}A}{192\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,A{b}^{4}}{384\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,A{b}^{4}}{128\,{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{B}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bb}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{b}^{2}}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{b}^{3}}{16\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{b}^{3}}{48\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,B{b}^{3}}{16\,a}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(B*x^2+A)/x^9,x)

[Out]

-1/8*A*(b*x^2+a)^(7/2)/a/x^8+1/48*A*b/a^2/x^6*(b*x^2+a)^(7/2)+1/192*A*b^2/a^3/x^
4*(b*x^2+a)^(7/2)+1/128*A*b^3/a^4/x^2*(b*x^2+a)^(7/2)-1/128*A*b^4/a^4*(b*x^2+a)^
(5/2)-5/384*A*b^4/a^3*(b*x^2+a)^(3/2)+5/128*A*b^4/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x
^2+a)^(1/2))/x)-5/128*A*b^4/a^2*(b*x^2+a)^(1/2)-1/6*B/a/x^6*(b*x^2+a)^(7/2)-1/24
*B*b/a^2/x^4*(b*x^2+a)^(7/2)-1/16*B*b^2/a^3/x^2*(b*x^2+a)^(7/2)+1/16*B*b^3/a^3*(
b*x^2+a)^(5/2)+5/48*B*b^3/a^2*(b*x^2+a)^(3/2)-5/16*B*b^3/a^(1/2)*ln((2*a+2*a^(1/
2)*(b*x^2+a)^(1/2))/x)+5/16*B*b^3/a*(b*x^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^2 + A)*(b*x^2 + a)^(5/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296411, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (8 \, B a b^{3} - A b^{4}\right )} x^{8} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (88 \, B a b^{2} + 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (104 \, B a^{2} b + 59 \, A a b^{2}\right )} x^{4} + 48 \, A a^{3} + 8 \,{\left (8 \, B a^{3} + 17 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{768 \, a^{\frac{3}{2}} x^{8}}, -\frac{15 \,{\left (8 \, B a b^{3} - A b^{4}\right )} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \,{\left (88 \, B a b^{2} + 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (104 \, B a^{2} b + 59 \, A a b^{2}\right )} x^{4} + 48 \, A a^{3} + 8 \,{\left (8 \, B a^{3} + 17 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{384 \, \sqrt{-a} a x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^9,x, algorithm="fricas")

[Out]

[-1/768*(15*(8*B*a*b^3 - A*b^4)*x^8*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 +
a)*a)/x^2) + 2*(3*(88*B*a*b^2 + 5*A*b^3)*x^6 + 2*(104*B*a^2*b + 59*A*a*b^2)*x^4
+ 48*A*a^3 + 8*(8*B*a^3 + 17*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(3/2)*x^
8), -1/384*(15*(8*B*a*b^3 - A*b^4)*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (3*(88
*B*a*b^2 + 5*A*b^3)*x^6 + 2*(104*B*a^2*b + 59*A*a*b^2)*x^4 + 48*A*a^3 + 8*(8*B*a
^3 + 17*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^8)]

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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**2+a)**(5/2)*(B*x**2+A)/x**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235339, size = 263, normalized size = 1.73 \[ \frac{\frac{15 \,{\left (8 \, B a b^{4} - A b^{5}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{264 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a b^{4} - 584 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 440 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{3} b^{4} - 120 \, \sqrt{b x^{2} + a} B a^{4} b^{4} + 15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A b^{5} + 73 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a b^{5} - 55 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} b^{5} + 15 \, \sqrt{b x^{2} + a} A a^{3} b^{5}}{a b^{4} x^{8}}}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^9,x, algorithm="giac")

[Out]

1/384*(15*(8*B*a*b^4 - A*b^5)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) - (2
64*(b*x^2 + a)^(7/2)*B*a*b^4 - 584*(b*x^2 + a)^(5/2)*B*a^2*b^4 + 440*(b*x^2 + a)
^(3/2)*B*a^3*b^4 - 120*sqrt(b*x^2 + a)*B*a^4*b^4 + 15*(b*x^2 + a)^(7/2)*A*b^5 +
73*(b*x^2 + a)^(5/2)*A*a*b^5 - 55*(b*x^2 + a)^(3/2)*A*a^2*b^5 + 15*sqrt(b*x^2 +
a)*A*a^3*b^5)/(a*b^4*x^8))/b