3.533 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=120 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{b \sqrt{a+b x^2} (A b-6 a B)}{16 a x^2}+\frac{\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]

[Out]

(b*(A*b - 6*a*B)*Sqrt[a + b*x^2])/(16*a*x^2) + ((A*b - 6*a*B)*(a + b*x^2)^(3/2))
/(24*a*x^4) - (A*(a + b*x^2)^(5/2))/(6*a*x^6) + (b^2*(A*b - 6*a*B)*ArcTanh[Sqrt[
a + b*x^2]/Sqrt[a]])/(16*a^(3/2))

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Rubi [A]  time = 0.25046, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{b \sqrt{a+b x^2} (A b-6 a B)}{16 a x^2}+\frac{\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]

[Out]

(b*(A*b - 6*a*B)*Sqrt[a + b*x^2])/(16*a*x^2) + ((A*b - 6*a*B)*(a + b*x^2)^(3/2))
/(24*a*x^4) - (A*(a + b*x^2)^(5/2))/(6*a*x^6) + (b^2*(A*b - 6*a*B)*ArcTanh[Sqrt[
a + b*x^2]/Sqrt[a]])/(16*a^(3/2))

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Rubi in Sympy [A]  time = 20.5432, size = 105, normalized size = 0.88 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{2}} \left (A b - 6 B a\right )}{16 a x^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - 6 B a\right )}{24 a x^{4}} + \frac{b^{2} \left (A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**7,x)

[Out]

-A*(a + b*x**2)**(5/2)/(6*a*x**6) + b*sqrt(a + b*x**2)*(A*b - 6*B*a)/(16*a*x**2)
+ (a + b*x**2)**(3/2)*(A*b - 6*B*a)/(24*a*x**4) + b**2*(A*b - 6*B*a)*atanh(sqrt
(a + b*x**2)/sqrt(a))/(16*a**(3/2))

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Mathematica [A]  time = 0.157332, size = 120, normalized size = 1. \[ \frac{b^2 (A b-6 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{16 a^{3/2}}-\frac{b^2 \log (x) (A b-6 a B)}{16 a^{3/2}}+\sqrt{a+b x^2} \left (\frac{-6 a B-7 A b}{24 x^4}-\frac{b (10 a B+A b)}{16 a x^2}-\frac{a A}{6 x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^7,x]

[Out]

(-(a*A)/(6*x^6) + (-7*A*b - 6*a*B)/(24*x^4) - (b*(A*b + 10*a*B))/(16*a*x^2))*Sqr
t[a + b*x^2] - (b^2*(A*b - 6*a*B)*Log[x])/(16*a^(3/2)) + (b^2*(A*b - 6*a*B)*Log[
a + Sqrt[a]*Sqrt[a + b*x^2]])/(16*a^(3/2))

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Maple [B]  time = 0.013, size = 233, normalized size = 1.9 \[ -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}A}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{3}}{48\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A{b}^{3}}{16\,{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{b}^{2}}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,B{b}^{2}}{8\,a}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^7,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258684, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (10 \, B a b + A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{3}{2}} x^{6}}, -\frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \,{\left (10 \, B a b + A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[-1/96*(3*(6*B*a*b^2 - A*b^3)*x^6*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a
)*a)/x^2) + 2*(3*(10*B*a*b + A*b^2)*x^4 + 8*A*a^2 + 2*(6*B*a^2 + 7*A*a*b)*x^2)*s
qrt(b*x^2 + a)*sqrt(a))/(a^(3/2)*x^6), -1/48*(3*(6*B*a*b^2 - A*b^3)*x^6*arctan(s
qrt(-a)/sqrt(b*x^2 + a)) + (3*(10*B*a*b + A*b^2)*x^4 + 8*A*a^2 + 2*(6*B*a^2 + 7*
A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^6)]

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Sympy [A]  time = 139.909, size = 253, normalized size = 2.11 \[ - \frac{A a^{2}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{11 A a \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{17 A b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{5}{2}}}{16 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{3}{2}}} - \frac{B a^{2}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B a \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{B b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**7,x)

[Out]

-A*a**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 11*A*a*sqrt(b)/(24*x**5*sqrt(a/(
b*x**2) + 1)) - 17*A*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - A*b**(5/2)/(16*a*
x*sqrt(a/(b*x**2) + 1)) + A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2)) - B*a*
*2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*a*sqrt(b)/(8*x**3*sqrt(a/(b*x**2)
+ 1)) - B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - B*b**(3/2)/(8*x*sqrt(a/(b*x**2)
+ 1)) - 3*B*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a))

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GIAC/XCAS [A]  time = 0.241533, size = 215, normalized size = 1.79 \[ \frac{\frac{3 \,{\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{30 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 18 \, \sqrt{b x^{2} + a} B a^{3} b^{3} + 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} - 3 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a b^{3} x^{6}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^7,x, algorithm="giac")

[Out]

1/48*(3*(6*B*a*b^3 - A*b^4)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) - (30*
(b*x^2 + a)^(5/2)*B*a*b^3 - 48*(b*x^2 + a)^(3/2)*B*a^2*b^3 + 18*sqrt(b*x^2 + a)*
B*a^3*b^3 + 3*(b*x^2 + a)^(5/2)*A*b^4 + 8*(b*x^2 + a)^(3/2)*A*a*b^4 - 3*sqrt(b*x
^2 + a)*A*a^2*b^4)/(a*b^3*x^6))/b