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

Optimal. Leaf size=152 \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{5 a x \sqrt{a+b x^2} (6 A b-7 a B)}{16 b^4}+\frac{5 x^3 \sqrt{a+b x^2} (6 A b-7 a B)}{24 b^3}-\frac{x^5 (6 A b-7 a B)}{6 b^2 \sqrt{a+b x^2}}+\frac{B x^7}{6 b \sqrt{a+b x^2}} \]

[Out]

-((6*A*b - 7*a*B)*x^5)/(6*b^2*Sqrt[a + b*x^2]) + (B*x^7)/(6*b*Sqrt[a + b*x^2]) -
(5*a*(6*A*b - 7*a*B)*x*Sqrt[a + b*x^2])/(16*b^4) + (5*(6*A*b - 7*a*B)*x^3*Sqrt[
a + b*x^2])/(24*b^3) + (5*a^2*(6*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(16*b^(9/2))

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Rubi [A]  time = 0.217056, antiderivative size = 152, 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 \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{5 a x \sqrt{a+b x^2} (6 A b-7 a B)}{16 b^4}+\frac{5 x^3 \sqrt{a+b x^2} (6 A b-7 a B)}{24 b^3}-\frac{x^5 (6 A b-7 a B)}{6 b^2 \sqrt{a+b x^2}}+\frac{B x^7}{6 b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-((6*A*b - 7*a*B)*x^5)/(6*b^2*Sqrt[a + b*x^2]) + (B*x^7)/(6*b*Sqrt[a + b*x^2]) -
(5*a*(6*A*b - 7*a*B)*x*Sqrt[a + b*x^2])/(16*b^4) + (5*(6*A*b - 7*a*B)*x^3*Sqrt[
a + b*x^2])/(24*b^3) + (5*a^2*(6*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(16*b^(9/2))

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Rubi in Sympy [A]  time = 23.2889, size = 148, normalized size = 0.97 \[ \frac{B x^{7}}{6 b \sqrt{a + b x^{2}}} + \frac{5 a^{2} \left (6 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{9}{2}}} - \frac{5 a x \sqrt{a + b x^{2}} \left (6 A b - 7 B a\right )}{16 b^{4}} - \frac{x^{5} \left (6 A b - 7 B a\right )}{6 b^{2} \sqrt{a + b x^{2}}} + \frac{5 x^{3} \sqrt{a + b x^{2}} \left (6 A b - 7 B a\right )}{24 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**7/(6*b*sqrt(a + b*x**2)) + 5*a**2*(6*A*b - 7*B*a)*atanh(sqrt(b)*x/sqrt(a +
b*x**2))/(16*b**(9/2)) - 5*a*x*sqrt(a + b*x**2)*(6*A*b - 7*B*a)/(16*b**4) - x**5
*(6*A*b - 7*B*a)/(6*b**2*sqrt(a + b*x**2)) + 5*x**3*sqrt(a + b*x**2)*(6*A*b - 7*
B*a)/(24*b**3)

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Mathematica [A]  time = 0.216914, size = 123, normalized size = 0.81 \[ \frac{5 a^2 (6 A b-7 a B) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{9/2}}+\frac{x \left (105 a^3 B+a^2 \left (35 b B x^2-90 A b\right )-2 a b^2 x^2 \left (15 A+7 B x^2\right )+4 b^3 x^4 \left (3 A+2 B x^2\right )\right )}{48 b^4 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(105*a^3*B + 4*b^3*x^4*(3*A + 2*B*x^2) - 2*a*b^2*x^2*(15*A + 7*B*x^2) + a^2*(
-90*A*b + 35*b*B*x^2)))/(48*b^4*Sqrt[a + b*x^2]) + (5*a^2*(6*A*b - 7*a*B)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/(16*b^(9/2))

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Maple [A]  time = 0.011, size = 185, normalized size = 1.2 \[{\frac{A{x}^{5}}{4\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,{a}^{2}Ax}{8\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{{x}^{7}B}{6\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{7\,Ba{x}^{5}}{24\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{2}B{x}^{3}}{48\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,B{a}^{3}x}{16\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*A*x^5/b/(b*x^2+a)^(1/2)-5/8*A*a/b^2*x^3/(b*x^2+a)^(1/2)-15/8*A*a^2/b^3*x/(b*
x^2+a)^(1/2)+15/8*A*a^2/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/6*B*x^7/b/(b*x^2
+a)^(1/2)-7/24*B*a/b^2*x^5/(b*x^2+a)^(1/2)+35/48*B*a^2/b^3*x^3/(b*x^2+a)^(1/2)+3
5/16*B*a^3/b^4*x/(b*x^2+a)^(1/2)-35/16*B*a^3/b^(9/2)*ln(x*b^(1/2)+(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)*x^6/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277213, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{3} x^{7} - 2 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{5} + 5 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{3} + 15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (7 \, B a^{4} - 6 \, A a^{3} b +{\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \,{\left (b^{5} x^{2} + a b^{4}\right )} \sqrt{b}}, \frac{{\left (8 \, B b^{3} x^{7} - 2 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{5} + 5 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{3} + 15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (7 \, B a^{4} - 6 \, A a^{3} b +{\left (7 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \,{\left (b^{5} x^{2} + a b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*B*b^3*x^7 - 2*(7*B*a*b^2 - 6*A*b^3)*x^5 + 5*(7*B*a^2*b - 6*A*a*b^2)*
x^3 + 15*(7*B*a^3 - 6*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(7*B*a^4 - 6*A*a^
3*b + (7*B*a^3*b - 6*A*a^2*b^2)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*
sqrt(b)))/((b^5*x^2 + a*b^4)*sqrt(b)), 1/48*((8*B*b^3*x^7 - 2*(7*B*a*b^2 - 6*A*b
^3)*x^5 + 5*(7*B*a^2*b - 6*A*a*b^2)*x^3 + 15*(7*B*a^3 - 6*A*a^2*b)*x)*sqrt(b*x^2
+ a)*sqrt(-b) - 15*(7*B*a^4 - 6*A*a^3*b + (7*B*a^3*b - 6*A*a^2*b^2)*x^2)*arctan
(sqrt(-b)*x/sqrt(b*x^2 + a)))/((b^5*x^2 + a*b^4)*sqrt(-b))]

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Sympy [A]  time = 61.5541, size = 233, normalized size = 1.53 \[ A \left (- \frac{15 a^{\frac{3}{2}} x}{8 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 \sqrt{a} x^{3}}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{x^{5}}{4 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + B \left (\frac{35 a^{\frac{5}{2}} x}{16 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} x^{3}}{48 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{7 \sqrt{a} x^{5}}{24 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{9}{2}}} + \frac{x^{7}}{6 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-15*a**(3/2)*x/(8*b**3*sqrt(1 + b*x**2/a)) - 5*sqrt(a)*x**3/(8*b**2*sqrt(1 +
b*x**2/a)) + 15*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(7/2)) + x**5/(4*sqrt(a)*b*s
qrt(1 + b*x**2/a))) + B*(35*a**(5/2)*x/(16*b**4*sqrt(1 + b*x**2/a)) + 35*a**(3/2
)*x**3/(48*b**3*sqrt(1 + b*x**2/a)) - 7*sqrt(a)*x**5/(24*b**2*sqrt(1 + b*x**2/a)
) - 35*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*b**(9/2)) + x**7/(6*sqrt(a)*b*sqrt(1 +
b*x**2/a)))

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GIAC/XCAS [A]  time = 0.238776, size = 184, normalized size = 1.21 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \, B x^{2}}{b} - \frac{7 \, B a b^{5} - 6 \, A b^{6}}{b^{7}}\right )} x^{2} + \frac{5 \,{\left (7 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )}}{b^{7}}\right )} x^{2} + \frac{15 \,{\left (7 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )}}{b^{7}}\right )} x}{48 \, \sqrt{b x^{2} + a}} + \frac{5 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*((2*(4*B*x^2/b - (7*B*a*b^5 - 6*A*b^6)/b^7)*x^2 + 5*(7*B*a^2*b^4 - 6*A*a*b^
5)/b^7)*x^2 + 15*(7*B*a^3*b^3 - 6*A*a^2*b^4)/b^7)*x/sqrt(b*x^2 + a) + 5/16*(7*B*
a^3 - 6*A*a^2*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)