3.535 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx\)

Optimal. Leaf size=156 \[ -\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}+\frac{\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}+\frac{b \sqrt{a+b x^2} (3 A b-8 a B)}{64 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8} \]

[Out]

(b*(3*A*b - 8*a*B)*Sqrt[a + b*x^2])/(64*a*x^4) + (b^2*(3*A*b - 8*a*B)*Sqrt[a + b
*x^2])/(128*a^2*x^2) + ((3*A*b - 8*a*B)*(a + b*x^2)^(3/2))/(48*a*x^6) - (A*(a +
b*x^2)^(5/2))/(8*a*x^8) - (b^3*(3*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])
/(128*a^(5/2))

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Rubi [A]  time = 0.309678, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}+\frac{\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}+\frac{b \sqrt{a+b x^2} (3 A b-8 a B)}{64 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]

[Out]

(b*(3*A*b - 8*a*B)*Sqrt[a + b*x^2])/(64*a*x^4) + (b^2*(3*A*b - 8*a*B)*Sqrt[a + b
*x^2])/(128*a^2*x^2) + ((3*A*b - 8*a*B)*(a + b*x^2)^(3/2))/(48*a*x^6) - (A*(a +
b*x^2)^(5/2))/(8*a*x^8) - (b^3*(3*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])
/(128*a^(5/2))

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Rubi in Sympy [A]  time = 25.6058, size = 143, normalized size = 0.92 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{8 a x^{8}} + \frac{b \sqrt{a + b x^{2}} \left (3 A b - 8 B a\right )}{64 a x^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 A b - 8 B a\right )}{48 a x^{6}} + \frac{b^{2} \sqrt{a + b x^{2}} \left (3 A b - 8 B a\right )}{128 a^{2} x^{2}} - \frac{b^{3} \left (3 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**9,x)

[Out]

-A*(a + b*x**2)**(5/2)/(8*a*x**8) + b*sqrt(a + b*x**2)*(3*A*b - 8*B*a)/(64*a*x**
4) + (a + b*x**2)**(3/2)*(3*A*b - 8*B*a)/(48*a*x**6) + b**2*sqrt(a + b*x**2)*(3*
A*b - 8*B*a)/(128*a**2*x**2) - b**3*(3*A*b - 8*B*a)*atanh(sqrt(a + b*x**2)/sqrt(
a))/(128*a**(5/2))

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Mathematica [A]  time = 0.200736, size = 145, normalized size = 0.93 \[ -\frac{b^3 (3 A b-8 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{128 a^{5/2}}+\frac{b^3 \log (x) (3 A b-8 a B)}{128 a^{5/2}}+\sqrt{a+b x^2} \left (-\frac{b^2 (8 a B-3 A b)}{128 a^2 x^2}+\frac{-8 a B-9 A b}{48 x^6}-\frac{b (56 a B+3 A b)}{192 a x^4}-\frac{a A}{8 x^8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^9,x]

[Out]

(-(a*A)/(8*x^8) + (-9*A*b - 8*a*B)/(48*x^6) - (b*(3*A*b + 56*a*B))/(192*a*x^4) -
(b^2*(-3*A*b + 8*a*B))/(128*a^2*x^2))*Sqrt[a + b*x^2] + (b^3*(3*A*b - 8*a*B)*Lo
g[x])/(128*a^(5/2)) - (b^3*(3*A*b - 8*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(12
8*a^(5/2))

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Maple [B]  time = 0.014, size = 275, normalized size = 1.8 \[ -{\frac{A}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{16\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}A}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,A{b}^{4}}{128\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{b}^{2}}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{b}^{3}}{48\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{b}^{3}}{16\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^9,x)

[Out]

-1/8*A*(b*x^2+a)^(5/2)/a/x^8+1/16*A*b/a^2/x^6*(b*x^2+a)^(5/2)-1/64*A*b^2/a^3/x^4
*(b*x^2+a)^(5/2)-1/128*A*b^3/a^4/x^2*(b*x^2+a)^(5/2)+1/128*A*b^4/a^4*(b*x^2+a)^(
3/2)-3/128*A*b^4/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/128*A*b^4/a^3*(
b*x^2+a)^(1/2)-1/6*B/a/x^6*(b*x^2+a)^(5/2)+1/24*B*b/a^2/x^4*(b*x^2+a)^(5/2)+1/48
*B*b^2/a^3/x^2*(b*x^2+a)^(5/2)-1/48*B*b^3/a^3*(b*x^2+a)^(3/2)+1/16*B*b^3/a^(3/2)
*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/16*B*b^3/a^2*(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)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296432, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} x^{8} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (8 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (56 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{4} + 48 \, A a^{3} + 8 \,{\left (8 \, B a^{3} + 9 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{768 \, a^{\frac{5}{2}} x^{8}}, \frac{3 \,{\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (8 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (56 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{4} + 48 \, A a^{3} + 8 \,{\left (8 \, B a^{3} + 9 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{384 \, \sqrt{-a} a^{2} x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^9,x, algorithm="fricas")

[Out]

[-1/768*(3*(8*B*a*b^3 - 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*(8*B*a*b^2 - 3*A*b^3)*x^6 + 2*(56*B*a^2*b + 3*A*a*b^2)*x^4 +
48*A*a^3 + 8*(8*B*a^3 + 9*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^8),
1/384*(3*(8*B*a*b^3 - 3*A*b^4)*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (3*(8*B*a
*b^2 - 3*A*b^3)*x^6 + 2*(56*B*a^2*b + 3*A*a*b^2)*x^4 + 48*A*a^3 + 8*(8*B*a^3 + 9
*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*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)**(3/2)*(B*x**2+A)/x**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252521, size = 262, normalized size = 1.68 \[ -\frac{\frac{3 \,{\left (8 \, B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{24 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a b^{4} + 40 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{2} b^{4} - 88 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{3} b^{4} + 24 \, \sqrt{b x^{2} + a} B a^{4} b^{4} - 9 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A b^{5} + 33 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a b^{5} + 33 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 9 \, \sqrt{b x^{2} + a} A a^{3} b^{5}}{a^{2} b^{4} x^{8}}}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^9,x, algorithm="giac")

[Out]

-1/384*(3*(8*B*a*b^4 - 3*A*b^5)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2)
+ (24*(b*x^2 + a)^(7/2)*B*a*b^4 + 40*(b*x^2 + a)^(5/2)*B*a^2*b^4 - 88*(b*x^2 + a
)^(3/2)*B*a^3*b^4 + 24*sqrt(b*x^2 + a)*B*a^4*b^4 - 9*(b*x^2 + a)^(7/2)*A*b^5 + 3
3*(b*x^2 + a)^(5/2)*A*a*b^5 + 33*(b*x^2 + a)^(3/2)*A*a^2*b^5 - 9*sqrt(b*x^2 + a)
*A*a^3*b^5)/(a^2*b^4*x^8))/b