Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}-\frac {2 a \left (a+b x^4\right )^{11/4}}{11 b^3}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^3}-\frac {2 a \left (a+b x^4\right )^{11/4}}{11 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{11} \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^2 (a+b x)^{3/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/4}}{b^2}-\frac {2 a (a+b x)^{7/4}}{b^2}+\frac {(a+b x)^{11/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac {a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}-\frac {2 a \left (a+b x^4\right )^{11/4}}{11 b^3}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 39, normalized size = 0.66 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (32 a^2-56 a b x^4+77 b^2 x^8\right )}{1155 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 36, normalized size = 0.61
method | result | size |
gosper | \(\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (77 b^{2} x^{8}-56 a b \,x^{4}+32 a^{2}\right )}{1155 b^{3}}\) | \(36\) |
trager | \(\frac {\left (77 b^{3} x^{12}+21 a \,b^{2} x^{8}-24 a^{2} b \,x^{4}+32 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 b^{3}}\) | \(47\) |
risch | \(\frac {\left (77 b^{3} x^{12}+21 a \,b^{2} x^{8}-24 a^{2} b \,x^{4}+32 a^{3}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{1155 b^{3}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {15}{4}}}{15 \, b^{3}} - \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a}{11 \, b^{3}} + \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{7 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 46, normalized size = 0.78 \begin {gather*} \frac {{\left (77 \, b^{3} x^{12} + 21 \, a b^{2} x^{8} - 24 \, a^{2} b x^{4} + 32 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.68, size = 87, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {32 a^{3} \left (a + b x^{4}\right )^{\frac {3}{4}}}{1155 b^{3}} - \frac {8 a^{2} x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{385 b^{2}} + \frac {a x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{55 b} + \frac {x^{12} \left (a + b x^{4}\right )^{\frac {3}{4}}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{4}} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (47) = 94\).
time = 1.36, size = 106, normalized size = 1.80 \begin {gather*} \frac {\frac {5 \, {\left (21 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} - 66 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a + 77 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2}\right )} a}{b^{2}} + \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} - 315 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a + 495 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2} - 385 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3}}{b^{2}}}{1155 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.08, size = 44, normalized size = 0.75 \begin {gather*} {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {x^{12}}{15}+\frac {32\,a^3}{1155\,b^3}+\frac {a\,x^8}{55\,b}-\frac {8\,a^2\,x^4}{385\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________