Optimal. Leaf size=111 \[ -a^2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )+\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}+a^2 c \sqrt {c+d x^2}-\frac {b \left (c+d x^2\right )^{5/2} (b c-2 a d)}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2} \]
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Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -a^2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )+\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}+a^2 c \sqrt {c+d x^2}-\frac {b \left (c+d x^2\right )^{5/2} (b c-2 a d)}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2} \end {gather*}
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 \left (c+d x^2\right )^{3/2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b (b c-2 a d) (c+d x)^{3/2}}{d}+\frac {a^2 (c+d x)^{3/2}}{x}+\frac {b^2 (c+d x)^{5/2}}{d}\right ) \, dx,x,x^2\right )\\ &=-\frac {b (b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2}+\frac {1}{2} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )\\ &=a^2 c \sqrt {c+d x^2}+\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2}+\frac {1}{2} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=a^2 c \sqrt {c+d x^2}+\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2}+\frac {\left (a^2 c^2\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 c \sqrt {c+d x^2}+\frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}-\frac {b (b c-2 a d) \left (c+d x^2\right )^{5/2}}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2}-a^2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 110, normalized size = 0.99 \begin {gather*} \frac {1}{3} a^2 \left (c+d x^2\right )^{3/2}+a^2 c \left (\sqrt {c+d x^2}-\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\right )+\frac {b \left (c+d x^2\right )^{5/2} (2 a d-b c)}{5 d^2}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 140, normalized size = 1.26 \begin {gather*} \frac {\sqrt {c+d x^2} \left (140 a^2 c d^2+35 a^2 d^3 x^2+42 a b c^2 d+84 a b c d^2 x^2+42 a b d^3 x^4-6 b^2 c^3+3 b^2 c^2 d x^2+24 b^2 c d^2 x^4+15 b^2 d^3 x^6\right )}{105 d^2}-a^2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 282, normalized size = 2.54 \begin {gather*} \left [\frac {105 \, a^{2} c^{\frac {3}{2}} d^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (15 \, b^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} + 42 \, a b c^{2} d + 140 \, a^{2} c d^{2} + 6 \, {\left (4 \, b^{2} c d^{2} + 7 \, a b d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{2} d + 84 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{210 \, d^{2}}, \frac {105 \, a^{2} \sqrt {-c} c d^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (15 \, b^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} + 42 \, a b c^{2} d + 140 \, a^{2} c d^{2} + 6 \, {\left (4 \, b^{2} c d^{2} + 7 \, a b d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{2} d + 84 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{105 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 121, normalized size = 1.09 \begin {gather*} \frac {a^{2} c^{2} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {15 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} d^{12} - 21 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c d^{12} + 42 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{13} + 35 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{14} + 105 \, \sqrt {d x^{2} + c} a^{2} c d^{14}}{105 \, d^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 1.04 \begin {gather*} -a^{2} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )+\sqrt {d \,x^{2}+c}\, a^{2} c +\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} x^{2}}{7 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2}}{3}+\frac {2 \left (d \,x^{2}+c \right )^{\frac {5}{2}} a b}{5 d}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c}{35 d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 103, normalized size = 0.93 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{2}}{7 \, d} - a^{2} c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} + \sqrt {d x^{2} + c} a^{2} c - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c}{35 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 191, normalized size = 1.72 \begin {gather*} {\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {{\left (a\,d-b\,c\right )}^2}{3\,d^2}-\frac {c\,\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d^2}-\frac {b^2\,c}{d^2}\right )}{3}\right )-\left (\frac {2\,b^2\,c-2\,a\,b\,d}{5\,d^2}-\frac {b^2\,c}{5\,d^2}\right )\,{\left (d\,x^2+c\right )}^{5/2}+\frac {b^2\,{\left (d\,x^2+c\right )}^{7/2}}{7\,d^2}+c\,\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 )+a^2\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 103.74, size = 109, normalized size = 0.98 \begin {gather*} \frac {a^{2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + a^{2} c \sqrt {c + d x^{2}} + \frac {a^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}{3} + \frac {b^{2} \left (c + d x^{2}\right )^{\frac {7}{2}}}{7 d^{2}} + \frac {\left (c + d x^{2}\right )^{\frac {5}{2}} \left (4 a b d - 2 b^{2} c\right )}{10 d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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