Optimal. Leaf size=147 \[ \frac {4 \left (\sqrt {a^2 x^2-b x}+a x\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}} \]
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Rubi [A] time = 0.23, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2121, 50, 63, 298, 205, 208} \begin {gather*} \frac {4 \left (\sqrt {a^2 x^2-b x}+a x\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 208
Rule 298
Rule 2121
Rubi steps
\begin {align*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^{3/4}}{-b+2 a x} \, dx,x,a x+\sqrt {-b x+a^2 x^2}\right )\\ &=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{x} (-b+2 a x)} \, dx,x,a x+\sqrt {-b x+a^2 x^2}\right )}{a}\\ &=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {x^2}{-b+2 a x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a}\\ &=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}-\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt {a} x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a^{3/2}}+\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt {a} x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a^{3/2}}\\ &=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}}\\ \end {align*}
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Mathematica [F] time = 1.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [F] time = 16.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.50, size = 268, normalized size = 1.82 \begin {gather*} -\frac {12 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {2 \, {\left (\left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} a^{2} b^{2} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} - \left (\frac {1}{8}\right )^{\frac {1}{4}} \sqrt {\sqrt {\frac {1}{2}} a^{3} b^{3} \sqrt {\frac {b^{3}}{a^{7}}} + \sqrt {a x + \sqrt {a^{2} x^{2} - b x}} b^{4}} a^{2} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right )}}{b^{3}}\right ) + 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 4 \, {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {3}{4}}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{4}}}{\sqrt {a^{2} x^{2}-b x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {3}{4}}}{\sqrt {a^{2} x^{2} - b x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b\,x}\right )}^{3/4}}{\sqrt {a^2\,x^2-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )^{\frac {3}{4}}}{\sqrt {x \left (a^{2} x - b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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