3.365 \(\int \frac{x^4 (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{\sqrt{a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}-\frac{3 a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2} \]

[Out]

-((x^3*(A + B*x))/(c*Sqrt[a + c*x^2])) + (4*B*x^2*Sqrt[a + c*x^2])/(3*c^2) - ((1
6*a*B - 9*A*c*x)*Sqrt[a + c*x^2])/(6*c^3) - (3*a*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(2*c^(5/2))

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Rubi [A]  time = 0.248883, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac{x^3 (A+B x)}{c \sqrt{a+c x^2}}-\frac{3 a A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}+\frac{4 B x^2 \sqrt{a+c x^2}}{3 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x))/(a + c*x^2)^(3/2),x]

[Out]

-((x^3*(A + B*x))/(c*Sqrt[a + c*x^2])) + (4*B*x^2*Sqrt[a + c*x^2])/(3*c^2) - ((1
6*a*B - 9*A*c*x)*Sqrt[a + c*x^2])/(6*c^3) - (3*a*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(2*c^(5/2))

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Rubi in Sympy [A]  time = 25.1338, size = 104, normalized size = 0.99 \[ - \frac{3 A a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{5}{2}}} + \frac{4 B x^{2} \sqrt{a + c x^{2}}}{3 c^{2}} - \frac{x^{3} \left (2 A + 2 B x\right )}{2 c \sqrt{a + c x^{2}}} - \frac{\sqrt{a + c x^{2}} \left (- 18 A c x + 32 B a\right )}{12 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(c*x**2+a)**(3/2),x)

[Out]

-3*A*a*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*c**(5/2)) + 4*B*x**2*sqrt(a + c*x**2
)/(3*c**2) - x**3*(2*A + 2*B*x)/(2*c*sqrt(a + c*x**2)) - sqrt(a + c*x**2)*(-18*A
*c*x + 32*B*a)/(12*c**3)

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Mathematica [A]  time = 0.160151, size = 85, normalized size = 0.81 \[ \frac{\frac{-16 a^2 B+a c x (9 A-8 B x)+c^2 x^3 (3 A+2 B x)}{\sqrt{a+c x^2}}-9 a A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x))/(a + c*x^2)^(3/2),x]

[Out]

((-16*a^2*B + a*c*x*(9*A - 8*B*x) + c^2*x^3*(3*A + 2*B*x))/Sqrt[a + c*x^2] - 9*a
*A*Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c^3)

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Maple [A]  time = 0.013, size = 115, normalized size = 1.1 \[{\frac{A{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,aAx}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,aA}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{x}^{4}B}{3\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{4\,aB{x}^{2}}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{8\,{a}^{2}B}{3\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)/(c*x^2+a)^(3/2),x)

[Out]

1/2*A*x^3/c/(c*x^2+a)^(1/2)+3/2*A*a/c^2*x/(c*x^2+a)^(1/2)-3/2*A*a/c^(5/2)*ln(c^(
1/2)*x+(c*x^2+a)^(1/2))+1/3*B*x^4/c/(c*x^2+a)^(1/2)-4/3*B*a/c^2*x^2/(c*x^2+a)^(1
/2)-8/3*B*a^2/c^3/(c*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 + A)*x^4/(c*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290484, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c} + 9 \,{\left (A a c^{2} x^{2} + A a^{2} c\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{12 \,{\left (c^{4} x^{2} + a c^{3}\right )} \sqrt{c}}, \frac{{\left (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c} - 9 \,{\left (A a c^{2} x^{2} + A a^{2} c\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{6 \,{\left (c^{4} x^{2} + a c^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(c*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(2*(2*B*c^2*x^4 + 3*A*c^2*x^3 - 8*B*a*c*x^2 + 9*A*a*c*x - 16*B*a^2)*sqrt(c
*x^2 + a)*sqrt(c) + 9*(A*a*c^2*x^2 + A*a^2*c)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x
^2 + a)*sqrt(c)))/((c^4*x^2 + a*c^3)*sqrt(c)), 1/6*((2*B*c^2*x^4 + 3*A*c^2*x^3 -
8*B*a*c*x^2 + 9*A*a*c*x - 16*B*a^2)*sqrt(c*x^2 + a)*sqrt(-c) - 9*(A*a*c^2*x^2 +
A*a^2*c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/((c^4*x^2 + a*c^3)*sqrt(-c))]

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Sympy [A]  time = 21.2012, size = 144, normalized size = 1.37 \[ A \left (\frac{3 \sqrt{a} x}{2 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + B \left (\begin{cases} - \frac{8 a^{2}}{3 c^{3} \sqrt{a + c x^{2}}} - \frac{4 a x^{2}}{3 c^{2} \sqrt{a + c x^{2}}} + \frac{x^{4}}{3 c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(c*x**2+a)**(3/2),x)

[Out]

A*(3*sqrt(a)*x/(2*c**2*sqrt(1 + c*x**2/a)) - 3*a*asinh(sqrt(c)*x/sqrt(a))/(2*c**
(5/2)) + x**3/(2*sqrt(a)*c*sqrt(1 + c*x**2/a))) + B*Piecewise((-8*a**2/(3*c**3*s
qrt(a + c*x**2)) - 4*a*x**2/(3*c**2*sqrt(a + c*x**2)) + x**4/(3*c*sqrt(a + c*x**
2)), Ne(c, 0)), (x**6/(6*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.278049, size = 112, normalized size = 1.07 \[ \frac{{\left ({\left ({\left (\frac{2 \, B x}{c} + \frac{3 \, A}{c}\right )} x - \frac{8 \, B a}{c^{2}}\right )} x + \frac{9 \, A a}{c^{2}}\right )} x - \frac{16 \, B a^{2}}{c^{3}}}{6 \, \sqrt{c x^{2} + a}} + \frac{3 \, A a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/6*((((2*B*x/c + 3*A/c)*x - 8*B*a/c^2)*x + 9*A*a/c^2)*x - 16*B*a^2/c^3)/sqrt(c*
x^2 + a) + 3/2*A*a*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)