Optimal. Leaf size=190 \[ -\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}+\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}}-\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-5 a d (12 b c-7 a d)}{48 c^3 x^2 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 193, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 89, 78, 51, 63, 208} \[ \frac {35 a^2 d^2-60 a b c d+24 b^2 c^2}{24 c^3 x^2 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 x^2}+\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^4 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} a (12 b c-7 a d)+3 b^2 c x}{x^3 (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c}\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {1}{48} \left (24 b^2-\frac {5 a d (12 b c-7 a d)}{c^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{24 c^3 x^2 \sqrt {c+d x^2}}+\frac {\left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{16 c^3}\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{24 c^3 x^2 \sqrt {c+d x^2}}-\frac {\left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^4 x^2}-\frac {\left (d \left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{32 c^4}\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{24 c^3 x^2 \sqrt {c+d x^2}}-\frac {\left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^4 x^2}-\frac {\left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{16 c^4}\\ &=-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}+\frac {24 b^2 c^2-60 a b c d+35 a^2 d^2}{24 c^3 x^2 \sqrt {c+d x^2}}-\frac {\left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \sqrt {c+d x^2}}{16 c^4 x^2}+\frac {d \left (24 b^2 c^2-60 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 92, normalized size = 0.48 \[ \frac {d x^6 \left (-35 a^2 d^2+60 a b c d-24 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {d x^2}{c}+1\right )+a c^2 \left (-4 a c+7 a d x^2-12 b c x^2\right )}{24 c^4 x^6 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 447, normalized size = 2.35 \[ \left [\frac {3 \, {\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + 8 \, a^{2} c^{4} + {\left (24 \, b^{2} c^{4} - 60 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, {\left (c^{5} d x^{8} + c^{6} x^{6}\right )}}, -\frac {3 \, {\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, {\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} + 8 \, a^{2} c^{4} + {\left (24 \, b^{2} c^{4} - 60 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (12 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, {\left (c^{5} d x^{8} + c^{6} x^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 267, normalized size = 1.41 \[ -\frac {{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{16 \, \sqrt {-c} c^{4}} - \frac {b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}}{\sqrt {d x^{2} + c} c^{4}} - \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d - 84 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{2} + 192 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 108 \, \sqrt {d x^{2} + c} a b c^{3} d^{2} + 57 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 136 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 87 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{3}}{48 \, c^{4} d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 281, normalized size = 1.48 \[ \frac {35 a^{2} d^{3} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{16 c^{\frac {9}{2}}}-\frac {15 a b \,d^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{4 c^{\frac {7}{2}}}+\frac {3 b^{2} d \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{2 c^{\frac {5}{2}}}-\frac {35 a^{2} d^{3}}{16 \sqrt {d \,x^{2}+c}\, c^{4}}+\frac {15 a b \,d^{2}}{4 \sqrt {d \,x^{2}+c}\, c^{3}}-\frac {3 b^{2} d}{2 \sqrt {d \,x^{2}+c}\, c^{2}}-\frac {35 a^{2} d^{2}}{48 \sqrt {d \,x^{2}+c}\, c^{3} x^{2}}+\frac {5 a b d}{4 \sqrt {d \,x^{2}+c}\, c^{2} x^{2}}-\frac {b^{2}}{2 \sqrt {d \,x^{2}+c}\, c \,x^{2}}+\frac {7 a^{2} d}{24 \sqrt {d \,x^{2}+c}\, c^{2} x^{4}}-\frac {a b}{2 \sqrt {d \,x^{2}+c}\, c \,x^{4}}-\frac {a^{2}}{6 \sqrt {d \,x^{2}+c}\, c \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 247, normalized size = 1.30 \[ \frac {3 \, b^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {5}{2}}} - \frac {15 \, a b d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{4 \, c^{\frac {7}{2}}} + \frac {35 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, c^{\frac {9}{2}}} - \frac {3 \, b^{2} d}{2 \, \sqrt {d x^{2} + c} c^{2}} + \frac {15 \, a b d^{2}}{4 \, \sqrt {d x^{2} + c} c^{3}} - \frac {35 \, a^{2} d^{3}}{16 \, \sqrt {d x^{2} + c} c^{4}} - \frac {b^{2}}{2 \, \sqrt {d x^{2} + c} c x^{2}} + \frac {5 \, a b d}{4 \, \sqrt {d x^{2} + c} c^{2} x^{2}} - \frac {35 \, a^{2} d^{2}}{48 \, \sqrt {d x^{2} + c} c^{3} x^{2}} - \frac {a b}{2 \, \sqrt {d x^{2} + c} c x^{4}} + \frac {7 \, a^{2} d}{24 \, \sqrt {d x^{2} + c} c^{2} x^{4}} - \frac {a^{2}}{6 \, \sqrt {d x^{2} + c} c x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 246, normalized size = 1.29 \[ \frac {d\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (35\,a^2\,d^2-60\,a\,b\,c\,d+24\,b^2\,c^2\right )}{16\,c^{9/2}}-\frac {\frac {a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d}{c}-\frac {\left (d\,x^2+c\right )\,\left (77\,a^2\,d^3-132\,a\,b\,c\,d^2+56\,b^2\,c^2\,d\right )}{16\,c^2}+\frac {{\left (d\,x^2+c\right )}^2\,\left (35\,a^2\,d^3-60\,a\,b\,c\,d^2+24\,b^2\,c^2\,d\right )}{6\,c^3}-\frac {{\left (d\,x^2+c\right )}^3\,\left (35\,a^2\,d^3-60\,a\,b\,c\,d^2+24\,b^2\,c^2\,d\right )}{16\,c^4}}{3\,c\,{\left (d\,x^2+c\right )}^{5/2}-{\left (d\,x^2+c\right )}^{7/2}+c^3\,\sqrt {d\,x^2+c}-3\,c^2\,{\left (d\,x^2+c\right )}^{3/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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