Optimal. Leaf size=153 \[ -\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}+\frac {105 a^3 \sqrt {b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac {105 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^{9/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2050, 2054,
212} \begin {gather*} -\frac {105 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{64 b^{9/2}}+\frac {105 a^3 \sqrt {a x+b x^{2/3}}}{64 b^4 x^{2/3}}-\frac {35 a^2 \sqrt {a x+b x^{2/3}}}{32 b^3 x}+\frac {7 a \sqrt {a x+b x^{2/3}}}{8 b^2 x^{4/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2050
Rule 2054
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}-\frac {(7 a) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}+\frac {\left (35 a^2\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{48 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}-\frac {\left (35 a^3\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{64 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}+\frac {105 a^3 \sqrt {b x^{2/3}+a x}}{64 b^4 x^{2/3}}+\frac {\left (35 a^4\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{128 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}+\frac {105 a^3 \sqrt {b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac {\left (105 a^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}+\frac {105 a^3 \sqrt {b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac {105 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 101, normalized size = 0.66 \begin {gather*} \frac {\sqrt {b x^{2/3}+a x} \left (-48 b^3+56 a b^2 \sqrt [3]{x}-70 a^2 b x^{2/3}+105 a^3 x\right )}{64 b^4 x^{5/3}}-\frac {105 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 126, normalized size = 0.82
method | result | size |
derivativedivides | \(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (48 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}}-56 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {7}{2}} a \,x^{\frac {1}{3}}+70 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {5}{2}} a^{2} x^{\frac {2}{3}}-105 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}} a^{3} x +105 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{4} b \,x^{\frac {4}{3}}\right )}{64 x \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {11}{2}}}\) | \(123\) |
default | \(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (105 x^{\frac {7}{3}} \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{4} b -56 x^{\frac {4}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {7}{2}} a -105 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}} a^{3}+70 x^{\frac {5}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {5}{2}} a^{2}+48 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}} x \right )}{64 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {11}{2}}}\) | \(126\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 109, normalized size = 0.71 \begin {gather*} \frac {\frac {105 \, a^{5} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {105 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{5} - 385 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{5} b + 511 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{5} b^{2} - 279 \, \sqrt {a x^{\frac {1}{3}} + b} a^{5} b^{3}}{a^{4} b^{4} x^{\frac {4}{3}}}}{64 \, a} \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 {1}{x^2\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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