Optimal. Leaf size=292 \[ -\frac {b d^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c^4}-\frac {\left (79 b^2+108 a b c+24 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{48 c^4 (b+a c) x^2}+\frac {(11 b+12 a c) d \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 c^4 x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{5/2}}{6 c^2 (b+a c) x^6}+\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{16 c^{9/2} (b+a c)^{3/2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1981,
1980, 474, 466, 1171, 396, 214} \begin {gather*} \frac {b d^3 \left (24 a^2 c^2+60 a b c+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{16 c^{9/2} (a c+b)^{3/2}}-\frac {d^2 \left (24 a^2 c^2+108 a b c+79 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{48 c^4 x^2 (a c+b)}-\frac {b d^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c^4}+\frac {d (12 a c+11 b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 c^4 x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{5/2}}{6 c^2 x^6 (a c+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 396
Rule 466
Rule 474
Rule 1171
Rule 1980
Rule 1981
Rule 1985
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a \left (c+d x^2\right )\right )^{3/2}}{x^7 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a c+a d x^2\right )^{3/2}}{x^7 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {(b+a c+a d x)^{3/2}}{x^4 (c+d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {7}{2} b (b+a c) d+3 a b d^2 x}{x^3 (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} b (b+a c) (35 b+32 a c) d^2+7 a b (b+a c) d^3 x}{x^2 (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {3}{8} b (b+a c) \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3+\frac {1}{4} a b (b+a c) (35 b+32 a c) d^4 x}{x (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {3 b^2 (b+a c) \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^4}{16 x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 b c^4 (b+a c)^2 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{32 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}+\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^{9/2} (b+a c)^{3/2} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 226, normalized size = 0.77 \begin {gather*} \frac {-\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (8 a^2 c^2 \left (c^3+d^3 x^6\right )+2 a b c \left (8 c^3-7 c^2 d x^2+16 c d^2 x^4+55 d^3 x^6\right )+b^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )}{(b+a c) x^6}+\frac {3 b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{(-b-a c)^{3/2}}}{48 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2604\) vs.
\(2(266)=532\).
time = 0.15, size = 2605, normalized size = 8.92
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} c^{2} d^{2} x^{4}+62 a c \,d^{2} b \,x^{4}-8 a^{2} c^{3} d \,x^{2}+57 b^{2} d^{2} x^{4}-30 a b \,c^{2} d \,x^{2}+8 a^{2} c^{4}-22 b^{2} c d \,x^{2}+16 a b \,c^{3}+8 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 c^{4} x^{6} \left (a c +b \right )}+\frac {\left (\frac {3 d^{3} b \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a^{2}}{4 c^{2} \left (a c +b \right ) \sqrt {c^{2} a +b c}}+\frac {15 d^{3} b^{2} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a}{8 c^{3} \left (a c +b \right ) \sqrt {c^{2} a +b c}}+\frac {35 d^{3} b^{3} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{32 c^{4} \left (a c +b \right ) \sqrt {c^{2} a +b c}}-\frac {d^{4} b \,x^{2} a^{2}}{c^{3} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{4} b^{2} x^{2} a}{c^{4} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{3} b \,a^{2}}{c^{2} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {2 d^{3} b^{2} a}{c^{3} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{3} b^{3}}{c^{4} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{a d \,x^{2}+a c +b}\) | \(775\) |
default | \(\text {Expression too large to display}\) | \(2605\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs.
\(2 (266) = 532\).
time = 0.54, size = 534, normalized size = 1.83 \begin {gather*} -\frac {{\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} d^{3} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{32 \, {\left (a c^{5} + b c^{4}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {b d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c^{4}} - \frac {3 \, {\left (8 \, a^{2} b c^{4} + 36 \, a b^{2} c^{3} + 29 \, b^{3} c^{2}\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{4} + 30 \, a^{2} b^{2} c^{3} + 41 \, a b^{3} c^{2} + 17 \, b^{4} c\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{4} + 44 \, a^{3} b^{2} c^{3} + 83 \, a^{2} b^{3} c^{2} + 66 \, a b^{4} c + 19 \, b^{5}\right )} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} c^{8} + 4 \, a^{3} b c^{7} + 6 \, a^{2} b^{2} c^{6} + 4 \, a b^{3} c^{5} + b^{4} c^{4} - \frac {{\left (a c^{8} + b c^{7}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {3 \, {\left (a^{2} c^{8} + 2 \, a b c^{7} + b^{2} c^{6}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (a^{3} c^{8} + 3 \, a^{2} b c^{7} + 3 \, a b^{2} c^{6} + b^{3} c^{5}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 733, normalized size = 2.51 \begin {gather*} \left [\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} \sqrt {a c^{2} + b c} d^{3} x^{6} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, {\left (a^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} \sqrt {-a c^{2} - b c} d^{3} x^{6} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, {\left (a^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}\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 (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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