Optimal. Leaf size=208 \[ \frac {315 a^{5/2} \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 277, 195, 217, 206} \[ \frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^4 \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 277
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2}}{x^2} \, 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^2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=-\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}}{x}+\frac {\left (9 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{7/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {\left (63 a b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{8 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {\left (105 a^2 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{16 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {\left (315 a^3 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \sqrt {a+b x^2} \, dx}{64 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {\left (315 a^4 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {\left (315 a^4 b c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {105}{64} a^2 b c x \sqrt {c \left (a+b x^2\right )^3}+\frac {315 a^3 b c x \sqrt {c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac {21}{16} a b c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{8} b c x \left (a+b x^2\right )^2 \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}}{x}+\frac {315 a^4 \sqrt {b} c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 65, normalized size = 0.31 \[ -\frac {a^4 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, _2F_1\left (-\frac {9}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \left (a+b x^2\right )^4 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 396, normalized size = 1.90 \[ \left [\frac {315 \, {\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt {b c} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c + 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt {b c} x}{b x^{2} + a}\right ) + 2 \, {\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{256 \, {\left (b x^{3} + a x\right )}}, -\frac {315 \, {\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt {-b c} x}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) - {\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{128 \, {\left (b x^{3} + a x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.54, size = 185, normalized size = 0.89 \[ \frac {1}{256} \, {\left (\frac {512 \, \sqrt {b c} a^{5} c \mathrm {sgn}\left (b x^{2} + a\right )}{{\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c}\right )}^{2} - a c} - 315 \, \sqrt {b c} a^{4} \log \left ({\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c}\right )}^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (325 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (105 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (2 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 11 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 215, normalized size = 1.03 \[ \frac {\left (\left (b \,x^{2}+a \right )^{3} c \right )^{\frac {3}{2}} \left (315 a^{4} b \,c^{3} x \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right )+315 \sqrt {b c \,x^{2}+a c}\, \sqrt {b c}\, a^{3} b \,c^{2} x^{2}+210 \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b c}\, a^{2} b c \,x^{2}+16 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, b^{2} x^{4}+56 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a b \,x^{2}-128 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a^{2}\right )}{128 \left (b \,x^{2}+a \right )^{3} \left (\left (b \,x^{2}+a \right ) c \right )^{\frac {3}{2}} \sqrt {b c}\, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________