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

Optimal. Leaf size=120 \[ \frac{3 B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{8 A c \sqrt{a+c x^2}}{3 a^3 x}-\frac{4 A \sqrt{a+c x^2}}{3 a^2 x^3}-\frac{3 B \sqrt{a+c x^2}}{2 a^2 x^2}+\frac{A+B x}{a x^3 \sqrt{a+c x^2}} \]

[Out]

(A + B*x)/(a*x^3*Sqrt[a + c*x^2]) - (4*A*Sqrt[a + c*x^2])/(3*a^2*x^3) - (3*B*Sqr
t[a + c*x^2])/(2*a^2*x^2) + (8*A*c*Sqrt[a + c*x^2])/(3*a^3*x) + (3*B*c*ArcTanh[S
qrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2))

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Rubi [A]  time = 0.34343, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{3 B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{8 A c \sqrt{a+c x^2}}{3 a^3 x}-\frac{4 A \sqrt{a+c x^2}}{3 a^2 x^3}-\frac{3 B \sqrt{a+c x^2}}{2 a^2 x^2}+\frac{A+B x}{a x^3 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(A + B*x)/(a*x^3*Sqrt[a + c*x^2]) - (4*A*Sqrt[a + c*x^2])/(3*a^2*x^3) - (3*B*Sqr
t[a + c*x^2])/(2*a^2*x^2) + (8*A*c*Sqrt[a + c*x^2])/(3*a^3*x) + (3*B*c*ArcTanh[S
qrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2))

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Rubi in Sympy [A]  time = 43.5733, size = 114, normalized size = 0.95 \[ - \frac{4 A \sqrt{a + c x^{2}}}{3 a^{2} x^{3}} + \frac{8 A c \sqrt{a + c x^{2}}}{3 a^{3} x} - \frac{3 B \sqrt{a + c x^{2}}}{2 a^{2} x^{2}} + \frac{3 B c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} + \frac{A + B x}{a x^{3} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*A*sqrt(a + c*x**2)/(3*a**2*x**3) + 8*A*c*sqrt(a + c*x**2)/(3*a**3*x) - 3*B*sq
rt(a + c*x**2)/(2*a**2*x**2) + 3*B*c*atanh(sqrt(a + c*x**2)/sqrt(a))/(2*a**(5/2)
) + (A + B*x)/(a*x**3*sqrt(a + c*x**2))

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Mathematica [A]  time = 0.236546, size = 100, normalized size = 0.83 \[ \frac{\frac{-a^2 (2 A+3 B x)+a c x^2 (8 A-9 B x)+16 A c^2 x^4}{x^3 \sqrt{a+c x^2}}+9 \sqrt{a} B c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-9 \sqrt{a} B c \log (x)}{6 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((16*A*c^2*x^4 + a*c*x^2*(8*A - 9*B*x) - a^2*(2*A + 3*B*x))/(x^3*Sqrt[a + c*x^2]
) - 9*Sqrt[a]*B*c*Log[x] + 9*Sqrt[a]*B*c*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(6*a^
3)

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Maple [A]  time = 0.014, size = 122, normalized size = 1. \[ -{\frac{A}{3\,a{x}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{4\,Ac}{3\,{a}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{8\,Ax{c}^{2}}{3\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{B}{2\,a{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,Bc}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,Bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*A/a/x^3/(c*x^2+a)^(1/2)+4/3*A/a^2*c/x/(c*x^2+a)^(1/2)+8/3*A/a^3*c^2*x/(c*x^
2+a)^(1/2)-1/2*B/a/x^2/(c*x^2+a)^(1/2)-3/2*B/a^2*c/(c*x^2+a)^(1/2)+3/2*B/a^(5/2)
*c*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)

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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)/((c*x^2 + a)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294551, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (16 \, A c^{2} x^{4} - 9 \, B a c x^{3} + 8 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a} + 9 \,{\left (B a c^{2} x^{5} + B a^{2} c x^{3}\right )} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} c x^{5} + a^{4} x^{3}\right )} \sqrt{a}}, \frac{{\left (16 \, A c^{2} x^{4} - 9 \, B a c x^{3} + 8 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a} + 9 \,{\left (B a c^{2} x^{5} + B a^{2} c x^{3}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{6 \,{\left (a^{3} c x^{5} + a^{4} x^{3}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(2*(16*A*c^2*x^4 - 9*B*a*c*x^3 + 8*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sqrt(c
*x^2 + a)*sqrt(a) + 9*(B*a*c^2*x^5 + B*a^2*c*x^3)*log(-((c*x^2 + 2*a)*sqrt(a) +
2*sqrt(c*x^2 + a)*a)/x^2))/((a^3*c*x^5 + a^4*x^3)*sqrt(a)), 1/6*((16*A*c^2*x^4 -
9*B*a*c*x^3 + 8*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sqrt(c*x^2 + a)*sqrt(-a) + 9*(
B*a*c^2*x^5 + B*a^2*c*x^3)*arctan(sqrt(-a)/sqrt(c*x^2 + a)))/((a^3*c*x^5 + a^4*x
^3)*sqrt(-a))]

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Sympy [A]  time = 34.3541, size = 311, normalized size = 2.59 \[ A \left (- \frac{a^{3} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac{3 a^{2} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac{12 a c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac{8 c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}}\right ) + B \left (- \frac{1}{2 a \sqrt{c} x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 \sqrt{c}}{2 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 a^{\frac{5}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-a**3*c**(9/2)*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4*x**2 + 6*a**4*c**5*x**4 + 3*
a**3*c**6*x**6) + 3*a**2*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4*x**2 +
6*a**4*c**5*x**4 + 3*a**3*c**6*x**6) + 12*a*c**(13/2)*x**4*sqrt(a/(c*x**2) + 1)
/(3*a**5*c**4*x**2 + 6*a**4*c**5*x**4 + 3*a**3*c**6*x**6) + 8*c**(15/2)*x**6*sqr
t(a/(c*x**2) + 1)/(3*a**5*c**4*x**2 + 6*a**4*c**5*x**4 + 3*a**3*c**6*x**6)) + B*
(-1/(2*a*sqrt(c)*x**3*sqrt(a/(c*x**2) + 1)) - 3*sqrt(c)/(2*a**2*x*sqrt(a/(c*x**2
) + 1)) + 3*c*asinh(sqrt(a)/(sqrt(c)*x))/(2*a**(5/2)))

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GIAC/XCAS [A]  time = 0.278688, size = 274, normalized size = 2.28 \[ \frac{\frac{A c^{2} x}{a^{3}} - \frac{B c}{a^{2}}}{\sqrt{c x^{2} + a}} - \frac{3 \, B c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B c - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A c^{\frac{3}{2}} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{2} c - 10 \, A a^{2} c^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c^2*x/a^3 - B*c/a^2)/sqrt(c*x^2 + a) - 3*B*c*arctan(-(sqrt(c)*x - sqrt(c*x^2
+ a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/3*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*c - 6*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*c^(3/2) + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A
*a*c^(3/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^2*c - 10*A*a^2*c^(3/2))/(((sqrt
(c)*x - sqrt(c*x^2 + a))^2 - a)^3*a^2)