Optimal. Leaf size=192 \[ \frac {a^4 c \sqrt {c \left (a+b x^2\right )^3}}{a+b x^2}+\frac {1}{3} a^3 c \sqrt {c \left (a+b x^2\right )^3}-\frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {1}{5} a^2 c \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{7} a c \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{9} c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.22, antiderivative size = 194, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 266, 50, 63, 208} \[ \frac {a^4 c \sqrt {c \left (a+b x^2\right )^3}}{a+b x^2}+\frac {1}{3} a^3 c \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{5} a^2 c \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}-\frac {a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\left (a+b x^2\right )^{3/2}}+\frac {1}{7} a c \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{9} c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2}}{x} \, dx &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{9} c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (a c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\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}+\frac {\left (a^2 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\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}+\frac {\left (a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\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}+\frac {\left (a^4 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{3} a^3 c \sqrt {c \left (a+b x^2\right )^3}+\frac {a^4 c \sqrt {c \left (a+b x^2\right )^3}}{a+b x^2}+\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}+\frac {\left (a^5 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{3} a^3 c \sqrt {c \left (a+b x^2\right )^3}+\frac {a^4 c \sqrt {c \left (a+b x^2\right )^3}}{a+b x^2}+\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}+\frac {\left (a^5 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b \left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{3} a^3 c \sqrt {c \left (a+b x^2\right )^3}+\frac {a^4 c \sqrt {c \left (a+b x^2\right )^3}}{a+b x^2}+\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}-\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}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 111, normalized size = 0.58 \[ \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {a+b x^2} \left (563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right )-315 a^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )}{315 \left (a+b x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 391, normalized size = 2.04 \[ \left [\frac {315 \, {\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt {a c} \log \left (-\frac {b^{2} c x^{4} + 3 \, a b c x^{2} + 2 \, a^{2} c - 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt {a c}}{b x^{4} + a x^{2}}\right ) + 2 \, {\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{630 \, {\left (b x^{2} + a\right )}}, \frac {315 \, {\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt {-a c} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt {-a c}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) + {\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{315 \, {\left (b x^{2} + a\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 185, normalized size = 0.96 \[ \frac {1}{315} \, {\left (\frac {315 \, a^{5} \arctan \left (\frac {\sqrt {b c x^{2} + a c}}{\sqrt {-a c}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {-a c}} + \frac {315 \, \sqrt {b c x^{2} + a c} a^{4} c^{44} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, {\left (b c x^{2} + a c\right )}^{\frac {3}{2}} a^{3} c^{43} \mathrm {sgn}\left (b x^{2} + a\right ) + 63 \, {\left (b c x^{2} + a c\right )}^{\frac {5}{2}} a^{2} c^{42} \mathrm {sgn}\left (b x^{2} + a\right ) + 45 \, {\left (b c x^{2} + a c\right )}^{\frac {7}{2}} a c^{41} \mathrm {sgn}\left (b x^{2} + a\right ) + 35 \, {\left (b c x^{2} + a c\right )}^{\frac {9}{2}} c^{40} \mathrm {sgn}\left (b x^{2} + a\right )}{c^{45}}\right )} c^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 221, normalized size = 1.15 \[ -\frac {\left (\left (b \,x^{2}+a \right )^{3} c \right )^{\frac {3}{2}} \left (315 a^{5} c^{3} \ln \left (\frac {2 a c +2 \sqrt {a c}\, \sqrt {b c \,x^{2}+a c}}{x}\right )-315 \sqrt {a c}\, \sqrt {b c \,x^{2}+a c}\, a^{4} c^{2}-35 \sqrt {a c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} b^{2} x^{4}-105 \sqrt {a c}\, \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{3} c -115 \sqrt {a c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a b \,x^{2}-189 \left (\left (b \,x^{2}+a \right ) c \right )^{\frac {5}{2}} \sqrt {a c}\, a^{2}+46 \sqrt {a c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{2}\right )}{315 \left (b \,x^{2}+a \right )^{3} \left (\left (b \,x^{2}+a \right ) c \right )^{\frac {3}{2}} \sqrt {a c}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (a + b x^{2}\right )^{3}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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