Optimal. Leaf size=136 \[ -\frac {256 b^4 \sqrt {a x^3+b x^4}}{315 a^5 x^2}+\frac {128 b^3 \sqrt {a x^3+b x^4}}{315 a^4 x^3}-\frac {32 b^2 \sqrt {a x^3+b x^4}}{105 a^3 x^4}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6} \]
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Rubi [A] time = 0.17, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2000} \[ -\frac {256 b^4 \sqrt {a x^3+b x^4}}{315 a^5 x^2}+\frac {128 b^3 \sqrt {a x^3+b x^4}}{315 a^4 x^3}-\frac {32 b^2 \sqrt {a x^3+b x^4}}{105 a^3 x^4}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6} \]
Antiderivative was successfully verified.
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Rule 2000
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a x^3+b x^4}} \, dx &=-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6}-\frac {(8 b) \int \frac {1}{x^3 \sqrt {a x^3+b x^4}} \, dx}{9 a}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}+\frac {\left (16 b^2\right ) \int \frac {1}{x^2 \sqrt {a x^3+b x^4}} \, dx}{21 a^2}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}-\frac {32 b^2 \sqrt {a x^3+b x^4}}{105 a^3 x^4}-\frac {\left (64 b^3\right ) \int \frac {1}{x \sqrt {a x^3+b x^4}} \, dx}{105 a^3}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}-\frac {32 b^2 \sqrt {a x^3+b x^4}}{105 a^3 x^4}+\frac {128 b^3 \sqrt {a x^3+b x^4}}{315 a^4 x^3}+\frac {\left (128 b^4\right ) \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx}{315 a^4}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{9 a x^6}+\frac {16 b \sqrt {a x^3+b x^4}}{63 a^2 x^5}-\frac {32 b^2 \sqrt {a x^3+b x^4}}{105 a^3 x^4}+\frac {128 b^3 \sqrt {a x^3+b x^4}}{315 a^4 x^3}-\frac {256 b^4 \sqrt {a x^3+b x^4}}{315 a^5 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 64, normalized size = 0.47 \[ -\frac {2 \sqrt {x^3 (a+b x)} \left (35 a^4-40 a^3 b x+48 a^2 b^2 x^2-64 a b^3 x^3+128 b^4 x^4\right )}{315 a^5 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 62, normalized size = 0.46 \[ -\frac {2 \, {\left (128 \, b^{4} x^{4} - 64 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a^{3} b x + 35 \, a^{4}\right )} \sqrt {b x^{4} + a x^{3}}}{315 \, a^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 71, normalized size = 0.52 \[ -\frac {2 \, {\left (35 \, {\left (b + \frac {a}{x}\right )}^{\frac {9}{2}} - 180 \, {\left (b + \frac {a}{x}\right )}^{\frac {7}{2}} b + 378 \, {\left (b + \frac {a}{x}\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (b + \frac {a}{x}\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {b + \frac {a}{x}} b^{4}\right )}}{315 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 0.50 \[ -\frac {2 \left (b x +a \right ) \left (128 b^{4} x^{4}-64 a \,b^{3} x^{3}+48 b^{2} x^{2} a^{2}-40 b x \,a^{3}+35 a^{4}\right )}{315 \sqrt {b \,x^{4}+a \,x^{3}}\, a^{5} x^{3}} \]
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 \[ \int \frac {1}{\sqrt {b x^{4} + a x^{3}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 116, normalized size = 0.85 \[ \frac {16\,b\,\sqrt {b\,x^4+a\,x^3}}{63\,a^2\,x^5}-\frac {2\,\sqrt {b\,x^4+a\,x^3}}{9\,a\,x^6}-\frac {32\,b^2\,\sqrt {b\,x^4+a\,x^3}}{105\,a^3\,x^4}+\frac {128\,b^3\,\sqrt {b\,x^4+a\,x^3}}{315\,a^4\,x^3}-\frac {256\,b^4\,\sqrt {b\,x^4+a\,x^3}}{315\,a^5\,x^2} \]
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 \[ \int \frac {1}{x^{4} \sqrt {x^{3} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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