Optimal. Leaf size=80 \[ \frac {a b d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {b d \left (a \sqrt {c+d x}+b c\right )}{2 c x} \]
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Rubi [A] time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {371, 1398, 821, 12, 639, 207} \begin {gather*} \frac {a b d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {b d \left (a \sqrt {c+d x}+b c\right )}{2 c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 371
Rule 639
Rule 821
Rule 1398
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx &=d^2 \operatorname {Subst}\left (\int \frac {\left (a+b \sqrt {x}\right )^2}{(-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x (a+b x)^2}{\left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {d^2 \operatorname {Subst}\left (\int -\frac {2 b c (a+b x)}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c}\\ &=-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {a+b x}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}-\frac {\left (a b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )}{2 c}\\ &=-\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\frac {a b d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [B] time = 0.35, size = 221, normalized size = 2.76 \begin {gather*} \frac {-\frac {2 \sqrt {c} \left (a^6 c+a^5 b \sqrt {c+d x} (2 c+d x)+a^4 b^2 \left (-c^2+2 c d x+3 d^2 x^2\right )-2 a^3 b^3 c \sqrt {c+d x} (2 c+d x)-a^2 b^4 c \left (c^2+4 c d x+2 d^2 x^2\right )+a b^5 c^2 \sqrt {c+d x} (2 c+d x)+b^6 c^2 (c+d x)^2\right )}{x^2 \left (a^2-b^2 c\right )^2}-a b d^2 \log \left (\sqrt {c}-\sqrt {c+d x}\right )+a b d^2 \log \left (\sqrt {c+d x}+\sqrt {c}\right )}{4 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 93, normalized size = 1.16 \begin {gather*} \frac {a^2 (-c)-a b (c+d x)^{3/2}-a b c \sqrt {c+d x}+b^2 c^2-2 b^2 c (c+d x)}{2 c x^2}+\frac {a b d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 181, normalized size = 2.26 \begin {gather*} \left [\frac {a b \sqrt {c} d^{2} x^{2} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 4 \, b^{2} c^{2} d x - 2 \, b^{2} c^{3} - 2 \, a^{2} c^{2} - 2 \, {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{4 \, c^{2} x^{2}}, -\frac {a b \sqrt {-c} d^{2} x^{2} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 2 \, b^{2} c^{2} d x + b^{2} c^{3} + a^{2} c^{2} + {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{2 \, c^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 105, normalized size = 1.31 \begin {gather*} -\frac {\frac {a b d^{3} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {2 \, {\left (d x + c\right )} b^{2} c d^{3} - b^{2} c^{2} d^{3} + {\left (d x + c\right )}^{\frac {3}{2}} a b d^{3} + \sqrt {d x + c} a b c d^{3} + a^{2} c d^{3}}{c d^{2} x^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 81, normalized size = 1.01 \begin {gather*} 4 \left (\frac {\arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}+\frac {-\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 c}-\frac {\sqrt {d x +c}}{8}}{d^{2} x^{2}}\right ) a b \,d^{2}+\left (-\frac {d}{x}-\frac {c}{2 x^{2}}\right ) b^{2}-\frac {a^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 113, normalized size = 1.41 \begin {gather*} -\frac {1}{4} \, {\left (\frac {a b \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )} b^{2} c - b^{2} c^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a b + \sqrt {d x + c} a b c + a^{2} c\right )}}{{\left (d x + c\right )}^{2} c - 2 \, {\left (d x + c\right )} c^{2} + c^{3}}\right )} d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.40, size = 80, normalized size = 1.00 \begin {gather*} \frac {a\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )}{2\,c^{3/2}}-\frac {b^2\,c}{2\,x^2}-\frac {b^2\,d}{x}-\frac {a\,b\,\sqrt {c+d\,x}}{2\,x^2}-\frac {a\,b\,{\left (c+d\,x\right )}^{3/2}}{2\,c\,x^2}-\frac {a^2}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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