Optimal. Leaf size=92 \[ a^2 \sqrt {c+d x^2}-a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )-\frac {b \left (c+d x^2\right )^{3/2} (b c-2 a d)}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 88, 50, 63, 208} \[ a^2 \sqrt {c+d x^2}-a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )-\frac {b \left (c+d x^2\right )^{3/2} (b c-2 a d)}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 \sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b (b c-2 a d) \sqrt {c+d x}}{d}+\frac {a^2 \sqrt {c+d x}}{x}+\frac {b^2 (c+d x)^{3/2}}{d}\right ) \, dx,x,x^2\right )\\ &=-\frac {b (b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=a^2 \sqrt {c+d x^2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {1}{2} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=a^2 \sqrt {c+d x^2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{d}\\ &=a^2 \sqrt {c+d x^2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{3/2}}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2}-a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 93, normalized size = 1.01 \[ a^2 \sqrt {c+d x^2}-a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )+\frac {b \left (c+d x^2\right )^{3/2} (2 a d-b c)}{3 d^2}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 207, normalized size = 2.25 \[ \left [\frac {15 \, a^{2} \sqrt {c} d^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{4} - 2 \, b^{2} c^{2} + 10 \, a b c d + 15 \, a^{2} d^{2} + {\left (b^{2} c d + 10 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, d^{2}}, \frac {15 \, a^{2} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, b^{2} d^{2} x^{4} - 2 \, b^{2} c^{2} + 10 \, a b c d + 15 \, a^{2} d^{2} + {\left (b^{2} c d + 10 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 101, normalized size = 1.10 \[ \frac {a^{2} c \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d^{8} - 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c d^{8} + 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{9} + 15 \, \sqrt {d x^{2} + c} a^{2} d^{10}}{15 \, d^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 100, normalized size = 1.09 \[ -a^{2} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} x^{2}}{5 d}+\sqrt {d \,x^{2}+c}\, a^{2}+\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b}{3 d}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c}{15 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 88, normalized size = 0.96 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x^{2}}{5 \, d} - a^{2} \sqrt {c} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) + \sqrt {d x^{2} + c} a^{2} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c}{15 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 135, normalized size = 1.47 \[ \sqrt {d\,x^2+c}\,\left (\frac {{\left (a\,d-b\,c\right )}^2}{d^2}-c\,\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d^2}-\frac {b^2\,c}{d^2}\right )\right )-\left (\frac {2\,b^2\,c-2\,a\,b\,d}{3\,d^2}-\frac {b^2\,c}{3\,d^2}\right )\,{\left (d\,x^2+c\right )}^{3/2}+\frac {b^2\,{\left (d\,x^2+c\right )}^{5/2}}{5\,d^2}+a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 72.46, size = 90, normalized size = 0.98 \[ \frac {a^{2} c \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + a^{2} \sqrt {c + d x^{2}} + \frac {b^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}{5 d^{2}} + \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (4 a b d - 2 b^{2} c\right )}{6 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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