Optimal. Leaf size=147 \[ -\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-4 A c) \sqrt {b x^2+c x^4}}{8 c^3}+\frac {(5 b B-4 A c) x^2 \sqrt {b x^2+c x^4}}{4 b c^2}+\frac {3 b (5 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{7/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2059, 802, 684,
654, 634, 212} \begin {gather*} \frac {3 b (5 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{7/2}}-\frac {3 \sqrt {b x^2+c x^4} (5 b B-4 A c)}{8 c^3}+\frac {x^2 \sqrt {b x^2+c x^4} (5 b B-4 A c)}{4 b c^2}-\frac {x^6 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 684
Rule 802
Rule 2059
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}+\frac {1}{2} \left (-\frac {4 A}{b}+\frac {5 B}{c}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}+\frac {(5 b B-4 A c) x^2 \sqrt {b x^2+c x^4}}{4 b c^2}-\frac {(3 (5 b B-4 A c)) \text {Subst}\left (\int \frac {x}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{8 c^2}\\ &=-\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-4 A c) \sqrt {b x^2+c x^4}}{8 c^3}+\frac {(5 b B-4 A c) x^2 \sqrt {b x^2+c x^4}}{4 b c^2}+\frac {(3 b (5 b B-4 A c)) \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{16 c^3}\\ &=-\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-4 A c) \sqrt {b x^2+c x^4}}{8 c^3}+\frac {(5 b B-4 A c) x^2 \sqrt {b x^2+c x^4}}{4 b c^2}+\frac {(3 b (5 b B-4 A c)) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^3}\\ &=-\frac {(b B-A c) x^6}{b c \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-4 A c) \sqrt {b x^2+c x^4}}{8 c^3}+\frac {(5 b B-4 A c) x^2 \sqrt {b x^2+c x^4}}{4 b c^2}+\frac {3 b (5 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 114, normalized size = 0.78 \begin {gather*} \frac {x \left (\sqrt {c} x \left (-15 b^2 B+b c \left (12 A-5 B x^2\right )+2 c^2 x^2 \left (2 A+B x^2\right )\right )-3 b (5 b B-4 A c) \sqrt {b+c x^2} \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{8 c^{7/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 140, normalized size = 0.95
method | result | size |
default | \(-\frac {x^{3} \left (c \,x^{2}+b \right ) \left (-2 B \,c^{\frac {7}{2}} x^{5}-4 A \,c^{\frac {7}{2}} x^{3}+5 B \,c^{\frac {5}{2}} b \,x^{3}-12 A \,c^{\frac {5}{2}} b x +15 B \,c^{\frac {3}{2}} b^{2} x +12 A \sqrt {c \,x^{2}+b}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b \,c^{2}-15 B \sqrt {c \,x^{2}+b}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{2} c \right )}{8 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} c^{\frac {9}{2}}}\) | \(140\) |
risch | \(\frac {x^{2} \left (2 B c \,x^{2}+4 A c -7 B b \right ) \left (c \,x^{2}+b \right )}{8 c^{3} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {\left (\frac {b x A}{c^{2} \sqrt {c \,x^{2}+b}}-\frac {b^{2} x B}{c^{3} \sqrt {c \,x^{2}+b}}-\frac {3 b \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) A}{2 c^{\frac {5}{2}}}+\frac {15 b^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) B}{8 c^{\frac {7}{2}}}\right ) x \sqrt {c \,x^{2}+b}}{\sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 187, normalized size = 1.27 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, x^{4}}{\sqrt {c x^{4} + b x^{2}} c} + \frac {6 \, b x^{2}}{\sqrt {c x^{4} + b x^{2}} c^{2}} - \frac {3 \, b \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {5}{2}}}\right )} A + \frac {1}{16} \, {\left (\frac {4 \, x^{6}}{\sqrt {c x^{4} + b x^{2}} c} - \frac {10 \, b x^{4}}{\sqrt {c x^{4} + b x^{2}} c^{2}} - \frac {30 \, b^{2} x^{2}}{\sqrt {c x^{4} + b x^{2}} c^{3}} + \frac {15 \, b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {7}{2}}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 289, normalized size = 1.97 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B b^{3} - 4 \, A b^{2} c + {\left (5 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (2 \, B c^{3} x^{4} - 15 \, B b^{2} c + 12 \, A b c^{2} - {\left (5 \, B b c^{2} - 4 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{16 \, {\left (c^{5} x^{2} + b c^{4}\right )}}, -\frac {3 \, {\left (5 \, B b^{3} - 4 \, A b^{2} c + {\left (5 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (2 \, B c^{3} x^{4} - 15 \, B b^{2} c + 12 \, A b c^{2} - {\left (5 \, B b c^{2} - 4 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{8 \, {\left (c^{5} x^{2} + b c^{4}\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 {x^{7} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 145, normalized size = 0.99 \begin {gather*} \frac {{\left (x^{2} {\left (\frac {2 \, B x^{2}}{c \mathrm {sgn}\left (x\right )} - \frac {5 \, B b c^{3} \mathrm {sgn}\left (x\right ) - 4 \, A c^{4} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} - \frac {3 \, {\left (5 \, B b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 4 \, A b c^{3} \mathrm {sgn}\left (x\right )\right )}}{c^{5}}\right )} x}{8 \, \sqrt {c x^{2} + b}} + \frac {3 \, {\left (5 \, B b^{2} \log \left ({\left | b \right |}\right ) - 4 \, A b c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {7}{2}}} - \frac {3 \, {\left (5 \, B b^{2} - 4 \, A b c\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{8 \, c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \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 = -0.01 \begin {gather*} \int \frac {x^7\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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