Optimal. Leaf size=75 \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+d x^2} (b c-2 a d)}{d^2}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^2} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 88, 63, 208} \begin {gather*} -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+d x^2} (b c-2 a d)}{d^2}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b (b c-2 a d)}{d \sqrt {c+d x}}+\frac {a^2}{x \sqrt {c+d x}}+\frac {b^2 \sqrt {c+d x}}{d}\right ) \, dx,x,x^2\right )\\ &=-\frac {b (b c-2 a d) \sqrt {c+d x^2}}{d^2}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {b (b c-2 a d) \sqrt {c+d x^2}}{d^2}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{d}\\ &=-\frac {b (b c-2 a d) \sqrt {c+d x^2}}{d^2}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 63, normalized size = 0.84 \begin {gather*} \frac {b \sqrt {c+d x^2} \left (6 a d-2 b c+b d x^2\right )}{3 d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 67, normalized size = 0.89 \begin {gather*} \frac {\sqrt {c+d x^2} \left (6 a b d-2 b^2 c+b^2 d x^2\right )}{3 d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.66, size = 157, normalized size = 2.09 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {c} d^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (b^{2} c d x^{2} - 2 \, b^{2} c^{2} + 6 \, a b c d\right )} \sqrt {d x^{2} + c}}{6 \, c d^{2}}, \frac {3 \, a^{2} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (b^{2} c d x^{2} - 2 \, b^{2} c^{2} + 6 \, a b c d\right )} \sqrt {d x^{2} + c}}{3 \, c d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 82, normalized size = 1.09 \begin {gather*} \frac {a^{2} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{4} - 3 \, \sqrt {d x^{2} + c} b^{2} c d^{4} + 6 \, \sqrt {d x^{2} + c} a b d^{5}}{3 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 1.16 \begin {gather*} \frac {\sqrt {d \,x^{2}+c}\, b^{2} x^{2}}{3 d}-\frac {a^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{\sqrt {c}}+\frac {2 \sqrt {d \,x^{2}+c}\, a b}{d}-\frac {2 \sqrt {d \,x^{2}+c}\, b^{2} c}{3 d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 75, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d x^{2} + c} b^{2} x^{2}}{3 \, d} - \frac {a^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{\sqrt {c}} - \frac {2 \, \sqrt {d x^{2} + c} b^{2} c}{3 \, d^{2}} + \frac {2 \, \sqrt {d x^{2} + c} a b}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 77, normalized size = 1.03 \begin {gather*} \frac {b^2\,{\left (d\,x^2+c\right )}^{3/2}}{3\,d^2}-\frac {a^2\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}-\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d^2}-\frac {b^2\,c}{d^2}\right )\,\sqrt {d\,x^2+c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 53.56, size = 76, normalized size = 1.01 \begin {gather*} \frac {a^{2} \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{c}} \sqrt {c + d x^{2}}} \right )}}{c \sqrt {- \frac {1}{c}}} + \frac {b^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}{3 d^{2}} + \frac {b \sqrt {c + d x^{2}} \left (2 a d - b c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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