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

Optimal. Leaf size=194 \[ -\frac{a^{9/2} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\left (a+b x^2\right )^{3/2}}+\frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

[Out]

(a^3*c*Sqrt[c*(a + b*x^2)^3])/3 + (a^4*c*Sqrt[c*(a + b*x^2)^3])/(a + b*x^2) + (a
^2*c*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/5 + (a*c*(a + b*x^2)^2*Sqrt[c*(a + b*x^2
)^3])/7 + (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/9 - (a^(9/2)*c*Sqrt[c*(a + b*x
^2)^3]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(a + b*x^2)^(3/2)

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Rubi [A]  time = 0.404341, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{a^{9/2} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\left (a+b x^2\right )^{3/2}}+\frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*c*Sqrt[c*(a + b*x^2)^3])/3 + (a^4*c*Sqrt[c*(a + b*x^2)^3])/(a + b*x^2) + (a
^2*c*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/5 + (a*c*(a + b*x^2)^2*Sqrt[c*(a + b*x^2
)^3])/7 + (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/9 - (a^(9/2)*c*Sqrt[c*(a + b*x
^2)^3]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(a + b*x^2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.156014, size = 122, normalized size = 0.63 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (-315 a^{9/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+315 a^{9/2} \log (x)+\sqrt{a+b x^2} \left (563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right )\right )}{315 \left (a+b x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((c*(a + b*x^2)^3)^(3/2)*(Sqrt[a + b*x^2]*(563*a^4 + 506*a^3*b*x^2 + 408*a^2*b^2
*x^4 + 185*a*b^3*x^6 + 35*b^4*x^8) + 315*a^(9/2)*Log[x] - 315*a^(9/2)*Log[a + Sq
rt[a]*Sqrt[a + b*x^2]]))/(315*(a + b*x^2)^(9/2))

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Maple [A]  time = 0.021, size = 221, normalized size = 1.1 \[ -{\frac{1}{315\,c \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( -35\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{4}{b}^{2}-115\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{2}ab+315\,{a}^{5}{c}^{3}\ln \left ( 2\,{\frac{\sqrt{ac}\sqrt{bc{x}^{2}+ac}+ac}{x}} \right ) +46\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{a}^{2}-105\,{a}^{3} \left ( bc{x}^{2}+ac \right ) ^{3/2}\sqrt{ac}c-315\,{a}^{4}{c}^{2}\sqrt{bc{x}^{2}+ac}\sqrt{ac}-189\,{a}^{2} \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{5/2}\sqrt{ac} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x^2+a)^3)^(3/2)/x,x)

[Out]

-1/315*(c*(b*x^2+a)^3)^(3/2)*(-35*(a*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*x^4*b^2-115*(a
*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*x^2*a*b+315*a^5*c^3*ln(2*((a*c)^(1/2)*(b*c*x^2+a*c
)^(1/2)+a*c)/x)+46*(a*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*a^2-105*a^3*(b*c*x^2+a*c)^(3/
2)*(a*c)^(1/2)*c-315*a^4*c^2*(b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)-189*a^2*(c*(b*x^2+a
))^(5/2)*(a*c)^(1/2))/(b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/(a*c)^(1/2)/c

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320006, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt{a c} \log \left (-\frac{b^{2} c x^{4} + 3 \, a b c x^{2} + 2 \, a^{2} c - 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{a c}}{b x^{4} + a x^{2}}\right ) + 2 \,{\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{630 \,{\left (b x^{2} + a\right )}}, -\frac{315 \,{\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt{-a c} \arctan \left (\frac{a b c x^{2} + a^{2} c}{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-a c}}\right ) -{\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{315 \,{\left (b x^{2} + a\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/630*(315*(a^4*b*c*x^2 + a^5*c)*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*(35*b^4*c*x^8 + 185*a*b^3*c*x^6 + 408*a^2*b^2*c*x^4 + 506*a^3*b*
c*x^2 + 563*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
^2 + a), -1/315*(315*(a^4*b*c*x^2 + a^5*c)*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))) - (35*b^4
*c*x^8 + 185*a*b^3*c*x^6 + 408*a^2*b^2*c*x^4 + 506*a^3*b*c*x^2 + 563*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^2 + a)]

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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,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275613, size = 194, normalized size = 1. \[ \frac{1}{315} \,{\left (\frac{315 \, a^{5} c \arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c}} + \frac{315 \, \sqrt{b c x^{2} + a c} a^{4} c^{36} + 105 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{35} + 63 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{34} + 45 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a c^{33} + 35 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}} c^{32}}{c^{36}}\right )} c{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(315*a^5*c*arctan(sqrt(b*c*x^2 + a*c)/sqrt(-a*c))/sqrt(-a*c) + (315*sqrt(b
*c*x^2 + a*c)*a^4*c^36 + 105*(b*c*x^2 + a*c)^(3/2)*a^3*c^35 + 63*(b*c*x^2 + a*c)
^(5/2)*a^2*c^34 + 45*(b*c*x^2 + a*c)^(7/2)*a*c^33 + 35*(b*c*x^2 + a*c)^(9/2)*c^3
2)/c^36)*c*sign(b*x^2 + a)