Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac {3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac {3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 80, 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 = {266, 43} \[ \frac {3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac {a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac {3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{15} \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x^3 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{5/4}}{b^3}+\frac {3 a^2 (a+b x)^{9/4}}{b^3}-\frac {3 a (a+b x)^{13/4}}{b^3}+\frac {(a+b x)^{17/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac {3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac {3 a \left (a+b x^4\right )^{17/4}}{17 b^4}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 50, normalized size = 0.62 \[ \frac {\left (a+b x^4\right )^{9/4} \left (-128 a^3+288 a^2 b x^4-468 a b^2 x^8+663 b^3 x^{12}\right )}{13923 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 68, normalized size = 0.85 \[ \frac {{\left (663 \, b^{5} x^{20} + 858 \, a b^{4} x^{16} + 15 \, a^{2} b^{3} x^{12} - 20 \, a^{3} b^{2} x^{8} + 32 \, a^{4} b x^{4} - 128 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{13923 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 57, normalized size = 0.71 \[ \frac {663 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} - 2457 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a + 3213 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{2} - 1547 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3}}{13923 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 47, normalized size = 0.59 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-663 b^{3} x^{12}+468 a \,b^{2} x^{8}-288 a^{2} b \,x^{4}+128 a^{3}\right )}{13923 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.31, size = 64, normalized size = 0.80 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {21}{4}}}{21 \, b^{4}} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a}{17 \, b^{4}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{2}}{13 \, b^{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3}}{9 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.14, size = 64, normalized size = 0.80 \[ {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {22\,a\,x^{16}}{357}+\frac {b\,x^{20}}{21}-\frac {128\,a^5}{13923\,b^4}+\frac {32\,a^4\,x^4}{13923\,b^3}-\frac {20\,a^3\,x^8}{13923\,b^2}+\frac {5\,a^2\,x^{12}}{4641\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 38.26, size = 134, normalized size = 1.68 \[ \begin {cases} - \frac {128 a^{5} \sqrt [4]{a + b x^{4}}}{13923 b^{4}} + \frac {32 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac {20 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{2}} + \frac {5 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b} + \frac {22 a x^{16} \sqrt [4]{a + b x^{4}}}{357} + \frac {b x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{4}} x^{16}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________