Optimal. Leaf size=90 \[ \frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {\sqrt {a+b x}}{3 a x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {\sqrt {a+b x}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x}}{3 a x^3}-\frac {(5 b) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{6 a}\\ &=-\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}+\frac {\left (5 b^2\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a^2}\\ &=-\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}-\frac {\left (5 b^3\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^3}\\ &=-\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a^3}\\ &=-\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.37 \[ \frac {2 b^3 \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {b x}{a}+1\right )}{a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 145, normalized size = 1.61 \[ \left [\frac {15 \, \sqrt {a} b^{3} x^{3} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{48 \, a^{4} x^{3}}, -\frac {15 \, \sqrt {-a} b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{24 \, a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 84, normalized size = 0.93 \[ -\frac {\frac {15 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} + 33 \, \sqrt {b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 1.00 \[ 2 \left (-\frac {5 \left (-\frac {3 \left (\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}-\frac {\sqrt {b x +a}}{2 a b x}\right )}{4 a}-\frac {\sqrt {b x +a}}{4 a \,b^{2} x^{2}}\right )}{6 a}-\frac {\sqrt {b x +a}}{6 a \,b^{3} x^{3}}\right ) b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 121, normalized size = 1.34 \[ -\frac {5 \, b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 33 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} a^{3} - 3 \, {\left (b x + a\right )}^{2} a^{4} + 3 \, {\left (b x + a\right )} a^{5} - a^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 69, normalized size = 0.77 \[ \frac {5\,{\left (a+b\,x\right )}^{3/2}}{3\,a^2\,x^3}-\frac {11\,\sqrt {a+b\,x}}{8\,a\,x^3}-\frac {5\,{\left (a+b\,x\right )}^{5/2}}{8\,a^3\,x^3}-\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.02, size = 129, normalized size = 1.43 \[ - \frac {1}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {\sqrt {b}}{12 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 b^{\frac {3}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 b^{\frac {5}{2}}}{8 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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