Optimal. Leaf size=190 \[ -\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 \sqrt {c+d x^2}}-\frac {a^2}{6 c x^6 \sqrt {c+d x^2}}-\frac {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}}-\frac {24 b^2 c^2-5 a d (12 b c-7 a d)}{48 c^3 x^2 \sqrt {c+d x^2}}+\frac {d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 c^{9/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {457, 91, 79, 44,
53, 65, 214} \begin {gather*} -\frac {35 a^2 d^2-60 a b c d+24 b^2 c^2}{48 c^3 x^2 \sqrt {c+d x^2}}-\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 {a (12 b c-7 a d)}{24 c^2 x^4 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 91
Rule 214
Rule 457
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} \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.27, size = 158, normalized size = 0.83 \begin {gather*} -\frac {24 b^2 c^2 x^4 \left (c+3 d x^2\right )+12 a b c x^2 \left (2 c^2-5 c d x^2-15 d^2 x^4\right )+a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )}{48 c^4 x^6 \sqrt {c+d 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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 284, normalized size = 1.49
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (57 a^{2} d^{2} x^{4}-84 a b c d \,x^{4}+24 b^{2} c^{2} x^{4}-22 a^{2} c d \,x^{2}+24 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{48 c^{4} x^{6}}-\frac {d^{3} a^{2}}{c^{4} \sqrt {d \,x^{2}+c}}+\frac {2 d^{2} a b}{c^{3} \sqrt {d \,x^{2}+c}}-\frac {d \,b^{2}}{c^{2} \sqrt {d \,x^{2}+c}}+\frac {35 d^{3} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a^{2}}{16 c^{\frac {9}{2}}}-\frac {15 d^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a b}{4 c^{\frac {7}{2}}}+\frac {3 d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b^{2}}{2 c^{\frac {5}{2}}}\) | \(235\) |
default | \(2 a b \left (-\frac {1}{4 c \,x^{4} \sqrt {d \,x^{2}+c}}-\frac {5 d \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )}{4 c}\right )+a^{2} \left (-\frac {1}{6 c \,x^{6} \sqrt {d \,x^{2}+c}}-\frac {7 d \left (-\frac {1}{4 c \,x^{4} \sqrt {d \,x^{2}+c}}-\frac {5 d \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )}{4 c}\right )}{6 c}\right )+b^{2} \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 247, normalized size = 1.30 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 447, normalized size = 2.35 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2}}{x^{7} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 267, normalized size = 1.41 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.71, size = 246, normalized size = 1.29 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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