Optimal. Leaf size=112 \[ -\frac {7 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{64 a^{11/4}}+\frac {7 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{64 a^{11/4}}+\frac {\left (32 a^2 x^2+4 a b x-7 b^2\right ) \sqrt [4]{a x^4+b x^3}}{96 a^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 196, normalized size of antiderivative = 1.75, number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2021, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} -\frac {7 b^3 x^{9/4} (a x+b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {7 b^3 x^{9/4} (a x+b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {7 b^2 \sqrt [4]{a x^4+b x^3}}{96 a^2}+\frac {b x \sqrt [4]{a x^4+b x^3}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x \sqrt [4]{b x^3+a x^4} \, dx &=\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {1}{12} b \int \frac {x^4}{\left (b x^3+a x^4\right )^{3/4}} \, dx\\ &=\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}-\frac {\left (7 b^2\right ) \int \frac {x^3}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{96 a}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {\left (7 b^3\right ) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{128 a^2}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {\left (7 b^3 x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{128 a^2 \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {\left (7 b^3 x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 a^2 \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {\left (7 b^3 x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32 a^2 \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}+\frac {\left (7 b^3 x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^3+a x^4\right )^{3/4}}-\frac {\left (7 b^3 x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {7 b^2 \sqrt [4]{b x^3+a x^4}}{96 a^2}+\frac {b x \sqrt [4]{b x^3+a x^4}}{24 a}+\frac {1}{3} x^2 \sqrt [4]{b x^3+a x^4}-\frac {7 b^3 x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^3+a x^4\right )^{3/4}}+\frac {7 b^3 x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 49, normalized size = 0.44 \begin {gather*} \frac {4 x^2 \sqrt [4]{x^3 (a x+b)} \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-\frac {a x}{b}\right )}{11 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.49, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-7 b^2+4 a b x+32 a^2 x^2\right ) \sqrt [4]{b x^3+a x^4}}{96 a^2}-\frac {7 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{11/4}}+\frac {7 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{11/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 253, normalized size = 2.26 \begin {gather*} -\frac {84 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{8} b^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {3}{4}} - a^{8} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {3}{4}} x \sqrt {\frac {a^{6} \sqrt {\frac {b^{12}}{a^{11}}} x^{2} + \sqrt {a x^{4} + b x^{3}} b^{6}}{x^{2}}}}{b^{12} x}\right ) - 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) + 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (32 \, a^{2} x^{2} + 4 \, a b x - 7 \, b^{2}\right )}}{384 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.38, size = 261, normalized size = 2.33 \begin {gather*} \frac {\frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} b^{4} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {8 \, {\left (7 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{4} - 18 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{4} - 21 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{4}\right )} x^{3}}{a^{2} b^{3}}}{768 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int x \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}\, dx\]
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*} \int {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x\,{d x} \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\,{\left (a\,x^4+b\,x^3\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt [4]{x^{3} \left (a x + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________