Optimal. Leaf size=149 \[ \frac {4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}-\frac {4 a^{7/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {281, 285, 327,
235, 233, 202} \begin {gather*} -\frac {4 a^{7/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^4}}+\frac {4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 233
Rule 235
Rule 281
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^9 \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^4 \left (a+b x^2\right )^{3/4} \, dx,x,x^2\right )\\ &=\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac {1}{26} (3 a) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}-\frac {a^2 \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{13 b}\\ &=-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{65 b^2}\\ &=-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac {\left (2 a^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{65 b^2 \sqrt [4]{a+b x^4}}\\ &=\frac {4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}-\frac {\left (2 a^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{65 b^2 \sqrt [4]{a+b x^4}}\\ &=\frac {4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac {a x^6 \left (a+b x^4\right )^{3/4}}{39 b}+\frac {1}{13} x^{10} \left (a+b x^4\right )^{3/4}-\frac {4 a^{7/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.39, size = 95, normalized size = 0.64 \begin {gather*} \frac {x^2 \left (a+b x^4\right )^{3/4} \left (\left (1+\frac {b x^4}{a}\right )^{3/4} \left (-2 a^2+a b x^4+3 b^2 x^8\right )+2 a^2 \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )\right )}{39 b^2 \left (1+\frac {b x^4}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{9} \left (b \,x^{4}+a \right )^{\frac {3}{4}}\, dx\]
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]
time = 0.08, size = 15, normalized size = 0.10 \begin {gather*} {\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{9}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.80, size = 29, normalized size = 0.19 \begin {gather*} \frac {a^{\frac {3}{4}} x^{10} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10} \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 = -0.01 \begin {gather*} \int x^9\,{\left (b\,x^4+a\right )}^{3/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________