Optimal. Leaf size=343 \[ -\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac {3 a^2 c^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}-\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.26, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {285, 327,
335, 338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 a^2 c^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{64 \sqrt {2} b^{7/4}}-\frac {3 a^2 c^{5/2} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{64 \sqrt {2} b^{7/4}}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 285
Rule 303
Rule 327
Rule 335
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int (c x)^{5/2} \sqrt [4]{a-b x^2} \, dx &=\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {1}{8} a \int \frac {(c x)^{5/2}}{\left (a-b x^2\right )^{3/4}} \, dx\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {\left (3 a^2 c^2\right ) \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx}{32 b}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {\left (3 a^2 c\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{16 b}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {\left (3 a^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{16 b}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac {\left (3 a^2 c\right ) \text {Subst}\left (\int \frac {c-\sqrt {b} x^2}{1+\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{32 b^{3/2}}+\frac {\left (3 a^2 c\right ) \text {Subst}\left (\int \frac {c+\sqrt {b} x^2}{1+\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{32 b^{3/2}}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {\left (3 a^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt [4]{b}}+2 x}{-\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {\left (3 a^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt [4]{b}}-2 x}{-\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {\left (3 a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 b^2}+\frac {\left (3 a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 b^2}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}+\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}-\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {\left (3 a^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}-\frac {\left (3 a^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}\\ &=-\frac {a c (c x)^{3/2} \sqrt [4]{a-b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a-b x^2}}{4 c}-\frac {3 a^2 c^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{32 \sqrt {2} b^{7/4}}+\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}-\frac {3 a^2 c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{64 \sqrt {2} b^{7/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.59, size = 177, normalized size = 0.52 \begin {gather*} \frac {(c x)^{5/2} \left (4 b^{3/4} x^{3/2} \sqrt [4]{a-b x^2} \left (-a+4 b x^2\right )+3 \sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}{-\sqrt {b} x+\sqrt {a-b x^2}}\right )-3 \sqrt {2} a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x+\sqrt {a-b x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}\right )\right )}{64 b^{7/4} x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 6.20, size = 48, normalized size = 0.14 \begin {gather*} \frac {\sqrt [4]{a} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________