Optimal. Leaf size=136 \[ -\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{32 a^{9/2}}+\frac {35 b^3 \sqrt {a+b \sqrt {x}}}{32 a^4 \sqrt {x}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {266, 51, 63, 208} \[ \frac {35 b^3 \sqrt {a+b \sqrt {x}}}{32 a^4 \sqrt {x}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}-\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{32 a^{9/2}}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {x}} x^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}-\frac {(7 b) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}+\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{24 a^2}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}-\frac {\left (35 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{32 a^3}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}+\frac {35 b^3 \sqrt {a+b \sqrt {x}}}{32 a^4 \sqrt {x}}+\frac {\left (35 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {x}\right )}{64 a^4}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}+\frac {35 b^3 \sqrt {a+b \sqrt {x}}}{32 a^4 \sqrt {x}}+\frac {\left (35 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {x}}\right )}{32 a^4}\\ &=-\frac {\sqrt {a+b \sqrt {x}}}{2 a x^2}+\frac {7 b \sqrt {a+b \sqrt {x}}}{12 a^2 x^{3/2}}-\frac {35 b^2 \sqrt {a+b \sqrt {x}}}{48 a^3 x}+\frac {35 b^3 \sqrt {a+b \sqrt {x}}}{32 a^4 \sqrt {x}}-\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {x}}}{\sqrt {a}}\right )}{32 a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 41, normalized size = 0.30 \[ -\frac {4 b^4 \sqrt {a+b \sqrt {x}} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};\frac {\sqrt {x} b}{a}+1\right )}{a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 184, normalized size = 1.35 \[ \left [\frac {105 \, \sqrt {a} b^{4} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b \sqrt {x} + a} \sqrt {a} \sqrt {x} + 2 \, a \sqrt {x}}{x}\right ) - 2 \, {\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \, {\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt {x}\right )} \sqrt {b \sqrt {x} + a}}{192 \, a^{5} x^{2}}, \frac {105 \, \sqrt {-a} b^{4} x^{2} \arctan \left (\frac {\sqrt {b \sqrt {x} + a} \sqrt {-a}}{a}\right ) - {\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \, {\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt {x}\right )} \sqrt {b \sqrt {x} + a}}{96 \, a^{5} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 109, normalized size = 0.80 \[ \frac {\frac {105 \, b^{5} \arctan \left (\frac {\sqrt {b \sqrt {x} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {105 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} b^{5} - 385 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a b^{5} + 511 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{2} b^{5} - 279 \, \sqrt {b \sqrt {x} + a} a^{3} b^{5}}{a^{4} b^{4} x^{2}}}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 124, normalized size = 0.91 \[ 4 \left (-\frac {7 \left (-\frac {5 \left (-\frac {3 \left (\frac {\arctanh \left (\frac {\sqrt {b \sqrt {x}+a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}-\frac {\sqrt {b \sqrt {x}+a}}{2 a b \sqrt {x}}\right )}{4 a}-\frac {\sqrt {b \sqrt {x}+a}}{4 a \,b^{2} x}\right )}{6 a}-\frac {\sqrt {b \sqrt {x}+a}}{6 a \,b^{3} x^{\frac {3}{2}}}\right )}{8 a}-\frac {\sqrt {b \sqrt {x}+a}}{8 a \,b^{4} x^{2}}\right ) b^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 166, normalized size = 1.22 \[ \frac {35 \, b^{4} \log \left (\frac {\sqrt {b \sqrt {x} + a} - \sqrt {a}}{\sqrt {b \sqrt {x} + a} + \sqrt {a}}\right )}{64 \, a^{\frac {9}{2}}} + \frac {105 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} b^{4} - 385 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a b^{4} + 511 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{2} b^{4} - 279 \, \sqrt {b \sqrt {x} + a} a^{3} b^{4}}{96 \, {\left ({\left (b \sqrt {x} + a\right )}^{4} a^{4} - 4 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5} + 6 \, {\left (b \sqrt {x} + a\right )}^{2} a^{6} - 4 \, {\left (b \sqrt {x} + a\right )} a^{7} + a^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 94, normalized size = 0.69 \[ \frac {511\,{\left (a+b\,\sqrt {x}\right )}^{3/2}}{96\,a^2\,x^2}-\frac {93\,\sqrt {a+b\,\sqrt {x}}}{32\,a\,x^2}-\frac {385\,{\left (a+b\,\sqrt {x}\right )}^{5/2}}{96\,a^3\,x^2}+\frac {35\,{\left (a+b\,\sqrt {x}\right )}^{7/2}}{32\,a^4\,x^2}+\frac {b^4\,\mathrm {atan}\left (\frac {\sqrt {a+b\,\sqrt {x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,35{}\mathrm {i}}{32\,a^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.85, size = 173, normalized size = 1.27 \[ - \frac {1}{2 \sqrt {b} x^{\frac {9}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {\sqrt {b}}{12 a x^{\frac {7}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {7 b^{\frac {3}{2}}}{48 a^{2} x^{\frac {5}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {35 b^{\frac {5}{2}}}{96 a^{3} x^{\frac {3}{4}} \sqrt {\frac {a}{b \sqrt {x}} + 1}} + \frac {35 b^{\frac {7}{2}}}{32 a^{4} \sqrt [4]{x} \sqrt {\frac {a}{b \sqrt {x}} + 1}} - \frac {35 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt [4]{x}} \right )}}{32 a^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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