Optimal. Leaf size=117 \[ \frac {2}{3} \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \tan ^{-1}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 141, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6720, 266, 50, 63, 298, 203, 206} \[ \frac {a^{3/4} c \sqrt {c \sqrt {a+b x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a+b x^2}}-\frac {a^{3/4} c \sqrt {c \sqrt {a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a+b x^2}}+\frac {2}{3} c \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {\left (a+b x^2\right )^{3/4}}{x} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/4}}{x} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac {2}{3} c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {\left (a c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac {2}{3} c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {\left (2 a c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{b \sqrt [4]{a+b x^2}}\\ &=\frac {2}{3} c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}-\frac {\left (a c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{\sqrt [4]{a+b x^2}}+\frac {\left (a c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{\sqrt [4]{a+b x^2}}\\ &=\frac {2}{3} c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}+\frac {a^{3/4} c \sqrt {c \sqrt {a+b x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a+b x^2}}-\frac {a^{3/4} c \sqrt {c \sqrt {a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 96, normalized size = 0.82 \[ \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (3 a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )-3 a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )+2 \left (a+b x^2\right )^{3/4}\right )}{3 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [A] time = 0.41, size = 190, normalized size = 1.62 \[ -\frac {1}{12} \, {\left (6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - 3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right ) + 3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right ) - 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}}\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.97, size = 118, normalized size = 1.01 \[ \frac {3 \, a c^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} c}}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}} + \frac {\log \left (\frac {\sqrt {\sqrt {b x^{2} + a} c} - \left (a c^{2}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b x^{2} + a} c} + \left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )} + 4 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} c^{2}}{6 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________