3.758 \(\int \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=204 \[ -\frac{9 a^{7/2} b c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \left (a+b x^2\right )^{3/2}}+\frac{9 a^3 b c \sqrt{c \left (a+b x^2\right )^3}}{2 \left (a+b x^2\right )}+\frac{3}{2} a^2 b c \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{10} a b c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}}{2 x^2}+\frac{9}{14} b c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3} \]

[Out]

(3*a^2*b*c*Sqrt[c*(a + b*x^2)^3])/2 + (9*a^3*b*c*Sqrt[c*(a + b*x^2)^3])/(2*(a +
b*x^2)) + (9*a*b*c*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/10 + (9*b*c*(a + b*x^2)^2*
Sqrt[c*(a + b*x^2)^3])/14 - (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/(2*x^2) - (9
*a^(7/2)*b*c*Sqrt[c*(a + b*x^2)^3]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*(a + b*x
^2)^(3/2))

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Rubi [A]  time = 0.424296, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{9 a^{7/2} b c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \left (a+b x^2\right )^{3/2}}+\frac{9 a^3 b c \sqrt{c \left (a+b x^2\right )^3}}{2 \left (a+b x^2\right )}+\frac{3}{2} a^2 b c \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{10} a b c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}}{2 x^2}+\frac{9}{14} b c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*b*c*Sqrt[c*(a + b*x^2)^3])/2 + (9*a^3*b*c*Sqrt[c*(a + b*x^2)^3])/(2*(a +
b*x^2)) + (9*a*b*c*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/10 + (9*b*c*(a + b*x^2)^2*
Sqrt[c*(a + b*x^2)^3])/14 - (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/(2*x^2) - (9
*a^(7/2)*b*c*Sqrt[c*(a + b*x^2)^3]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*(a + b*x
^2)^(3/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c \left (a + b x^{2}\right )^{3}\right )^{\frac{3}{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*(b*x**2+a)**3)**(3/2)/x**3,x)

[Out]

Integral((c*(a + b*x**2)**3)**(3/2)/x**3, x)

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Mathematica [A]  time = 0.15492, size = 133, normalized size = 0.65 \[ -\frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (-315 a^{7/2} b x^2 \log (x)+315 a^{7/2} b x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a+b x^2} \left (35 a^4-388 a^3 b x^2-156 a^2 b^2 x^4-58 a b^3 x^6-10 b^4 x^8\right )\right )}{70 x^2 \left (a+b x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(a + b*x^2)^3)^(3/2)*(Sqrt[a + b*x^2]*(35*a^4 - 388*a^3*b*x^2 - 156*a^2*b^2
*x^4 - 58*a*b^3*x^6 - 10*b^4*x^8) - 315*a^(7/2)*b*x^2*Log[x] + 315*a^(7/2)*b*x^2
*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]))/(70*x^2*(a + b*x^2)^(9/2))

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Maple [A]  time = 0.018, size = 238, normalized size = 1.2 \[ -{\frac{1}{70\, \left ( b{x}^{2}+a \right ) ^{3}{x}^{2}c} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( -10\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{4}{b}^{2}+315\,{a}^{4}b{c}^{3}\ln \left ( 2\,{\frac{\sqrt{ac}\sqrt{bc{x}^{2}+ac}+ac}{x}} \right ){x}^{2}+4\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{2}ab-105\,{a}^{2}b \left ( bc{x}^{2}+ac \right ) ^{3/2}{x}^{2}c\sqrt{ac}-315\,{a}^{3}b{c}^{2}\sqrt{bc{x}^{2}+ac}{x}^{2}\sqrt{ac}-42\,ab \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{5/2}{x}^{2}\sqrt{ac}+35\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{a}^{2} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/70*(c*(b*x^2+a)^3)^(3/2)*(-10*(a*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*x^4*b^2+315*a^4
*b*c^3*ln(2*((a*c)^(1/2)*(b*c*x^2+a*c)^(1/2)+a*c)/x)*x^2+4*(a*c)^(1/2)*(b*c*x^2+
a*c)^(5/2)*x^2*a*b-105*a^2*b*(b*c*x^2+a*c)^(3/2)*x^2*c*(a*c)^(1/2)-315*a^3*b*c^2
*(b*c*x^2+a*c)^(1/2)*x^2*(a*c)^(1/2)-42*a*b*(c*(b*x^2+a))^(5/2)*x^2*(a*c)^(1/2)+
35*(a*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*a^2)/(b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/x^2/c/(a
*c)^(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)^3*c)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300757, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a^{3} b^{2} c x^{4} + a^{4} b c x^{2}\right )} \sqrt{a c} \log \left (-\frac{b^{2} c x^{4} + 3 \, a b c x^{2} + 2 \, a^{2} c - 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{a c}}{b x^{4} + a x^{2}}\right ) + 2 \,{\left (10 \, b^{4} c x^{8} + 58 \, a b^{3} c x^{6} + 156 \, a^{2} b^{2} c x^{4} + 388 \, a^{3} b c x^{2} - 35 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{140 \,{\left (b x^{4} + a x^{2}\right )}}, -\frac{315 \,{\left (a^{3} b^{2} c x^{4} + a^{4} b c x^{2}\right )} \sqrt{-a c} \arctan \left (\frac{a b c x^{2} + a^{2} c}{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-a c}}\right ) -{\left (10 \, b^{4} c x^{8} + 58 \, a b^{3} c x^{6} + 156 \, a^{2} b^{2} c x^{4} + 388 \, a^{3} b c x^{2} - 35 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{70 \,{\left (b x^{4} + a x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/140*(315*(a^3*b^2*c*x^4 + a^4*b*c*x^2)*sqrt(a*c)*log(-(b^2*c*x^4 + 3*a*b*c*x^
2 + 2*a^2*c - 2*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(a*c
))/(b*x^4 + a*x^2)) + 2*(10*b^4*c*x^8 + 58*a*b^3*c*x^6 + 156*a^2*b^2*c*x^4 + 388
*a^3*b*c*x^2 - 35*a^4*c)*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)
)/(b*x^4 + a*x^2), -1/70*(315*(a^3*b^2*c*x^4 + a^4*b*c*x^2)*sqrt(-a*c)*arctan((a
*b*c*x^2 + a^2*c)/(sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(
-a*c))) - (10*b^4*c*x^8 + 58*a*b^3*c*x^6 + 156*a^2*b^2*c*x^4 + 388*a^3*b*c*x^2 -
35*a^4*c)*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b*x^4 + a*x
^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*(b*x**2+a)**3)**(3/2)/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276985, size = 204, normalized size = 1. \[ \frac{1}{70} \,{\left (\frac{315 \, a^{4} c^{2} \arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c}} - \frac{35 \, \sqrt{b c x^{2} + a c} a^{4} c}{b x^{2}} + \frac{2 \,{\left (140 \, \sqrt{b c x^{2} + a c} a^{3} c^{15} + 35 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{14} + 14 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a c^{13} + 5 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} c^{12}\right )}}{c^{14}}\right )} b{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/70*(315*a^4*c^2*arctan(sqrt(b*c*x^2 + a*c)/sqrt(-a*c))/sqrt(-a*c) - 35*sqrt(b*
c*x^2 + a*c)*a^4*c/(b*x^2) + 2*(140*sqrt(b*c*x^2 + a*c)*a^3*c^15 + 35*(b*c*x^2 +
a*c)^(3/2)*a^2*c^14 + 14*(b*c*x^2 + a*c)^(5/2)*a*c^13 + 5*(b*c*x^2 + a*c)^(7/2)
*c^12)/c^14)*b*sign(b*x^2 + a)