Optimal. Leaf size=78 \[ -\frac {a^3 \sqrt [4]{a+b x^4}}{b^4}+\frac {3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}-\frac {a \left (a+b x^4\right )^{9/4}}{3 b^4}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 78, 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^3 \sqrt [4]{a+b x^4}}{b^4}+\frac {3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^4}-\frac {a \left (a+b x^4\right )^{9/4}}{3 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^3}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{3/4}}+\frac {3 a^2 \sqrt [4]{a+b x}}{b^3}-\frac {3 a (a+b x)^{5/4}}{b^3}+\frac {(a+b x)^{9/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^3 \sqrt [4]{a+b x^4}}{b^4}+\frac {3 a^2 \left (a+b x^4\right )^{5/4}}{5 b^4}-\frac {a \left (a+b x^4\right )^{9/4}}{3 b^4}+\frac {\left (a+b x^4\right )^{13/4}}{13 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 50, normalized size = 0.64 \begin {gather*} \frac {\sqrt [4]{a+b x^4} \left (-128 a^3+32 a^2 b x^4-20 a b^2 x^8+15 b^3 x^{12}\right )}{195 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 47, normalized size = 0.60
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-15 b^{3} x^{12}+20 a \,b^{2} x^{8}-32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(47\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-15 b^{3} x^{12}+20 a \,b^{2} x^{8}-32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(47\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-15 b^{3} x^{12}+20 a \,b^{2} x^{8}-32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 64, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} a}{3 \, b^{4}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{5 \, b^{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 46, normalized size = 0.59 \begin {gather*} \frac {{\left (15 \, b^{3} x^{12} - 20 \, a b^{2} x^{8} + 32 \, a^{2} b x^{4} - 128 \, a^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{195 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.78, size = 92, normalized size = 1.18 \begin {gather*} \begin {cases} - \frac {128 a^{3} \sqrt [4]{a + b x^{4}}}{195 b^{4}} + \frac {32 a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{195 b^{3}} - \frac {4 a x^{8} \sqrt [4]{a + b x^{4}}}{39 b^{2}} + \frac {x^{12} \sqrt [4]{a + b x^{4}}}{13 b} & \text {for}\: b \neq 0 \\\frac {x^{16}}{16 a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.01, size = 61, normalized size = 0.78 \begin {gather*} -\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{b^{4}} + \frac {15 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} - 65 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a + 117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{195 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.18, size = 48, normalized size = 0.62 \begin {gather*} -{\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {128\,a^3}{195\,b^4}-\frac {x^{12}}{13\,b}+\frac {4\,a\,x^8}{39\,b^2}-\frac {32\,a^2\,x^4}{195\,b^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________