Optimal. Leaf size=96 \[ -\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270}
\begin {gather*} -\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx &=-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}+\frac {(12 b) \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx}{17 a}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}+\frac {\left (96 b^2\right ) \int \frac {\sqrt [4]{a-b x^4}}{x^{10}} \, dx}{221 a^2}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {\left (128 b^3\right ) \int \frac {\sqrt [4]{a-b x^4}}{x^6} \, dx}{663 a^3}\\ &=-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 54, normalized size = 0.56 \begin {gather*} \frac {\left (a-b x^4\right )^{5/4} \left (-195 a^3-180 a^2 b x^4-160 a b^2 x^8-128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 51, normalized size = 0.53
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (128 b^{3} x^{12}+160 a \,b^{2} x^{8}+180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\) | \(51\) |
trager | \(-\frac {\left (-128 x^{16} b^{4}-32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}-15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\) | \(62\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-128 x^{16} b^{4}-32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}-15 a^{3} b \,x^{4}+195 a^{4}\right )}{3315 x^{17} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}} a^{4}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 73, normalized size = 0.76 \begin {gather*} -\frac {\frac {663 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}{x^{5}} + \frac {1105 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} + \frac {765 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} + \frac {195 \, {\left (-b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{3315 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 61, normalized size = 0.64 \begin {gather*} \frac {{\left (128 \, b^{4} x^{16} + 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} + 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, a^{4} x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.65, size = 1756, normalized size = 18.29 \begin {gather*} \text {Too large to display} \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 [B]
time = 1.76, size = 98, normalized size = 1.02 \begin {gather*} \frac {b\,{\left (a-b\,x^4\right )}^{1/4}}{221\,a\,x^{13}}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{17\,x^{17}}+\frac {128\,b^4\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^4\,x}+\frac {32\,b^3\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^3\,x^5}+\frac {4\,b^2\,{\left (a-b\,x^4\right )}^{1/4}}{663\,a^2\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________