Optimal. Leaf size=99 \[ \frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{16 a^{7/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{16 a^{7/4}}+\frac {(4 a x+b) \sqrt [4]{a x^4+b x^3}}{8 a} \]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 168, normalized size of antiderivative = 1.70, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2004, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {3 b^2 x^{9/4} (a x+b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {3 b^2 x^{9/4} (a x+b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {1}{2} x \sqrt [4]{a x^4+b x^3}+\frac {b \sqrt [4]{a x^4+b x^3}}{8 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2004
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [4]{b x^3+a x^4} \, dx &=\frac {1}{2} x \sqrt [4]{b x^3+a x^4}+\frac {1}{8} b \int \frac {x^3}{\left (b x^3+a x^4\right )^{3/4}} \, dx\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b^2\right ) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{32 a}\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b^2 x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{32 a \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b^2 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 )}{8 a \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b^2 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 )}{8 a \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b^2 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 )}{16 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}+\frac {\left (3 b^2 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 )}{16 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {1}{2} x \sqrt [4]{b x^3+a x^4}+\frac {3 b^2 x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}-\frac {3 b^2 x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 47, normalized size = 0.47 \begin {gather*} \frac {4 x \sqrt [4]{x^3 (a x+b)} \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-\frac {a x}{b}\right )}{7 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.42, size = 99, normalized size = 1.00 \begin {gather*} \frac {(b+4 a x) \sqrt [4]{b x^3+a x^4}}{8 a}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{7/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.95, size = 234, normalized size = 2.36 \begin {gather*} \frac {12 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{5} b^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {3}{4}} - a^{5} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {3}{4}} x \sqrt {\frac {a^{4} \sqrt {\frac {b^{8}}{a^{7}}} x^{2} + \sqrt {a x^{4} + b x^{3}} b^{4}}{x^{2}}}}{b^{8} x}\right ) - 3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, {\left (a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2}\right )}}{x}\right ) + 3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2}\right )}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (4 \, a x + b\right )}}{32 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 243, normalized size = 2.45 \begin {gather*} \frac {\frac {6 \, \sqrt {2} b^{3} \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 )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {6 \, \sqrt {2} b^{3} \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 )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {3 \, \sqrt {2} b^{3} \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} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \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^{2}} + \frac {8 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{3} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{3}\right )} x^{2}}{a b^{2}}}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}\, dx\]
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}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.86, size = 38, normalized size = 0.38 \begin {gather*} \frac {4\,x\,{\left (a\,x^4+b\,x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {7}{4};\ \frac {11}{4};\ -\frac {a\,x}{b}\right )}{7\,{\left (\frac {a\,x}{b}+1\right )}^{1/4}} \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 \sqrt [4]{a x^{4} + b x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________