3.587 \(\int \frac{x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}-\frac{x (4 A b-7 a B)}{3 b^3 \sqrt{a+b x^2}}+\frac{a x (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B x \sqrt{a+b x^2}}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.232127, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{7/2}}-\frac{x (4 A b-7 a B)}{3 b^3 \sqrt{a+b x^2}}+\frac{a x (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{B x \sqrt{a+b x^2}}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.9397, size = 105, normalized size = 0.92 \[ \frac{B x \sqrt{a + b x^{2}}}{2 b^{3}} + \frac{a x \left (A b - B a\right )}{3 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (4 A b - 7 B a\right )}{3 b^{3} \sqrt{a + b x^{2}}} + \frac{\left (2 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

B*x*sqrt(a + b*x**2)/(2*b**3) + a*x*(A*b - B*a)/(3*b**3*(a + b*x**2)**(3/2)) - x
*(4*A*b - 7*B*a)/(3*b**3*sqrt(a + b*x**2)) + (2*A*b - 5*B*a)*atanh(sqrt(b)*x/sqr
t(a + b*x**2))/(2*b**(7/2))

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Mathematica [A]  time = 0.130437, size = 98, normalized size = 0.86 \[ \frac{x \left (15 a^2 B+a \left (20 b B x^2-6 A b\right )+b^2 x^2 \left (3 B x^2-8 A\right )\right )}{6 b^3 \left (a+b x^2\right )^{3/2}}+\frac{(2 A b-5 a B) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(x*(15*a^2*B + b^2*x^2*(-8*A + 3*B*x^2) + a*(-6*A*b + 20*b*B*x^2)))/(6*b^3*(a +
b*x^2)^(3/2)) + ((2*A*b - 5*a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*b^(7/2))

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Maple [A]  time = 0.012, size = 134, normalized size = 1.2 \[ -{\frac{A{x}^{3}}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Ax}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{A\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{{x}^{5}B}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,Ba{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,Bxa}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,Ba}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2575, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, B b^{2} x^{5} + 4 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{12 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{b}}, \frac{{\left (3 \, B b^{2} x^{5} + 4 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^4/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(2*(3*B*b^2*x^5 + 4*(5*B*a*b - 2*A*b^2)*x^3 + 3*(5*B*a^2 - 2*A*a*b)*x)*sqr
t(b*x^2 + a)*sqrt(b) - 3*((5*B*a*b^2 - 2*A*b^3)*x^4 + 5*B*a^3 - 2*A*a^2*b + 2*(5
*B*a^2*b - 2*A*a*b^2)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/
((b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(b)), 1/6*((3*B*b^2*x^5 + 4*(5*B*a*b - 2*
A*b^2)*x^3 + 3*(5*B*a^2 - 2*A*a*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 3*((5*B*a*b^2 -
2*A*b^3)*x^4 + 5*B*a^3 - 2*A*a^2*b + 2*(5*B*a^2*b - 2*A*a*b^2)*x^2)*arctan(sqrt
(-b)*x/sqrt(b*x^2 + a)))/((b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(-b))]

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Sympy [A]  time = 65.9155, size = 675, normalized size = 5.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

A*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b*
*(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) + 3*
a**(37/2)*b**12*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b*
*(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 3*
a**19*b**(23/2)*x/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29
/2)*x**2*sqrt(1 + b*x**2/a)) - 4*a**18*b**(25/2)*x**3/(3*a**(39/2)*b**(27/2)*sqr
t(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a))) + B*(-15*a**(8
1/2)*b**22*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*b**(51/2)*sq
rt(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)*s
qrt(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)))

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GIAC/XCAS [A]  time = 0.235055, size = 151, normalized size = 1.32 \[ \frac{{\left ({\left (\frac{3 \, B x^{2}}{b} + \frac{4 \,{\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )}}{a b^{5}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )}}{a b^{5}}\right )} x}{6 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, B a - 2 \, A b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^4/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

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