3.585 \(\int \frac{x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}+\frac{5 x \sqrt{a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac{5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt{a+b x^2}}-\frac{x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]

[Out]

-((4*A*b - 7*a*B)*x^5)/(12*b^2*(a + b*x^2)^(3/2)) + (B*x^7)/(4*b*(a + b*x^2)^(3/
2)) - (5*(4*A*b - 7*a*B)*x^3)/(12*b^3*Sqrt[a + b*x^2]) + (5*(4*A*b - 7*a*B)*x*Sq
rt[a + b*x^2])/(8*b^4) - (5*a*(4*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(8*b^(9/2))

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Rubi [A]  time = 0.205493, antiderivative size = 149, 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 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}+\frac{5 x \sqrt{a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac{5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt{a+b x^2}}-\frac{x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

-((4*A*b - 7*a*B)*x^5)/(12*b^2*(a + b*x^2)^(3/2)) + (B*x^7)/(4*b*(a + b*x^2)^(3/
2)) - (5*(4*A*b - 7*a*B)*x^3)/(12*b^3*Sqrt[a + b*x^2]) + (5*(4*A*b - 7*a*B)*x*Sq
rt[a + b*x^2])/(8*b^4) - (5*a*(4*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 23.2873, size = 144, normalized size = 0.97 \[ \frac{B x^{7}}{4 b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{5 a \left (4 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{9}{2}}} - \frac{x^{5} \left (4 A b - 7 B a\right )}{12 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3} \left (4 A b - 7 B a\right )}{12 b^{3} \sqrt{a + b x^{2}}} + \frac{5 x \sqrt{a + b x^{2}} \left (4 A b - 7 B a\right )}{8 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

B*x**7/(4*b*(a + b*x**2)**(3/2)) - 5*a*(4*A*b - 7*B*a)*atanh(sqrt(b)*x/sqrt(a +
b*x**2))/(8*b**(9/2)) - x**5*(4*A*b - 7*B*a)/(12*b**2*(a + b*x**2)**(3/2)) - 5*x
**3*(4*A*b - 7*B*a)/(12*b**3*sqrt(a + b*x**2)) + 5*x*sqrt(a + b*x**2)*(4*A*b - 7
*B*a)/(8*b**4)

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Mathematica [A]  time = 0.175981, size = 120, normalized size = 0.81 \[ \frac{-105 a^3 B x+20 a^2 b x \left (3 A-7 B x^2\right )+a b^2 x^3 \left (80 A-21 B x^2\right )+6 b^3 x^5 \left (2 A+B x^2\right )}{24 b^4 \left (a+b x^2\right )^{3/2}}+\frac{5 a (7 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(-105*a^3*B*x + a*b^2*x^3*(80*A - 21*B*x^2) + 20*a^2*b*x*(3*A - 7*B*x^2) + 6*b^3
*x^5*(2*A + B*x^2))/(24*b^4*(a + b*x^2)^(3/2)) + (5*a*(-4*A*b + 7*a*B)*Log[b*x +
Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(9/2))

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Maple [A]  time = 0.011, size = 181, normalized size = 1.2 \[{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,aA{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,aAx}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{{x}^{7}B}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,Ba{x}^{5}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}B{x}^{3}}{24\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,Bx{a}^{2}}{8\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29066, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (6 \, B b^{3} x^{7} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{5} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{48 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{b}}, \frac{{\left (6 \, B b^{3} x^{7} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{5} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{24 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(2*(6*B*b^3*x^7 - 3*(7*B*a*b^2 - 4*A*b^3)*x^5 - 20*(7*B*a^2*b - 4*A*a*b^2)
*x^3 - 15*(7*B*a^3 - 4*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(7*B*a^4 - 4*A*a
^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*log(2*sq
rt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*s
qrt(b)), 1/24*((6*B*b^3*x^7 - 3*(7*B*a*b^2 - 4*A*b^3)*x^5 - 20*(7*B*a^2*b - 4*A*
a*b^2)*x^3 - 15*(7*B*a^3 - 4*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 15*(7*B*a^4
- 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*a
rctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*sqrt(-b))]

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Sympy [A]  time = 108.445, size = 804, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

A*(-15*a**(81/2)*b**22*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*
b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) -
15*a**(79/2)*b**23*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)
*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) +
15*a**40*b**(45/2)*x/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b*
*(53/2)*x**2*sqrt(1 + b*x**2/a)) + 20*a**39*b**(47/2)*x**3/(6*a**(79/2)*b**(51/2
)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 3*a**38*
b**(49/2)*x**5/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)
*x**2*sqrt(1 + b*x**2/a))) + B*(105*a**(157/2)*b**41*sqrt(1 + b*x**2/a)*asinh(sq
rt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**
(93/2)*x**2*sqrt(1 + b*x**2/a)) + 105*a**(155/2)*b**42*x**2*sqrt(1 + b*x**2/a)*a
sinh(sqrt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151
/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) - 105*a**78*b**(83/2)*x/(24*a**(153/2)*b*
*(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) -
140*a**77*b**(85/2)*x**3/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(15
1/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) - 21*a**76*b**(87/2)*x**5/(24*a**(153/2)
*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a))
+ 6*a**75*b**(89/2)*x**7/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(1
51/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.232782, size = 200, normalized size = 1.34 \[ \frac{{\left ({\left (3 \,{\left (\frac{2 \, B x^{2}}{b} - \frac{7 \, B a^{2} b^{5} - 4 \, A a b^{6}}{a b^{7}}\right )} x^{2} - \frac{20 \,{\left (7 \, B a^{3} b^{4} - 4 \, A a^{2} b^{5}\right )}}{a b^{7}}\right )} x^{2} - \frac{15 \,{\left (7 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )}}{a b^{7}}\right )} x}{24 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (7 \, B a^{2} - 4 \, A a b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^6/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*B*x^2/b - (7*B*a^2*b^5 - 4*A*a*b^6)/(a*b^7))*x^2 - 20*(7*B*a^3*b^4 -
4*A*a^2*b^5)/(a*b^7))*x^2 - 15*(7*B*a^4*b^3 - 4*A*a^3*b^4)/(a*b^7))*x/(b*x^2 +
a)^(3/2) - 5/8*(7*B*a^2 - 4*A*a*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)