Optimal. Leaf size=409 \[ -\frac {x \left (c+d x^2\right )}{a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {4 x \left (b+a c+a d x^2\right )}{3 a^2 d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(8 b+a c) x \left (b+a c+a d x^2\right )}{3 a^3 d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {c} (8 b+a c) \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^3 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (4 b+a c) \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^2 (b+a c) d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.29, antiderivative size = 409, 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, 1986,
478, 542, 545, 429, 506, 422} \begin {gather*} \frac {\sqrt {c} (a c+8 b) \left (a c+a d x^2+b\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^3 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {x (a c+8 b) \left (a c+a d x^2+b\right )}{3 a^3 d \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {c^{3/2} (a c+4 b) \left (a c+a d x^2+b\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^2 d^{3/2} (a c+b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {4 x \left (a c+a d x^2+b\right )}{3 a^2 d \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {x \left (c+d x^2\right )}{a d \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 478
Rule 506
Rule 542
Rule 545
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\sqrt {c+d x^2} \left (c+4 d x^2\right )}{\sqrt {b+a c+a d x^2}} \, dx}{a d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {4 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {-c (4 b+a c) d-(8 b+a c) d^2 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a^2 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {4 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left ((8 b+a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c (4 b+a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {4 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^3 d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c^{3/2} (4 b+a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^2 (b+a c) d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c (8 b+a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^3 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {4 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a^3 d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {c} (8 b+a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^3 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {c^{3/2} (4 b+a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^2 (b+a c) d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.39, size = 255, normalized size = 0.62 \begin {gather*} \frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} x \left (c+d x^2\right ) \left (4 b+a c+a d x^2\right )+i c (8 b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )-4 i b c \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{3 a^2 d \sqrt {\frac {a d}{b+a c}} \left (b+a \left (c+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 667, normalized size = 1.63
method | result | size |
default | \(\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,d^{2} x^{5}+2 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a c d \,x^{3}+3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b d \,x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,c^{2} x +4 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c -8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c +3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, b c x +\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c x \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 a^{2} d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a d \,x^{2}+a c +b \right )}\) | \(667\) |
risch | \(\frac {x \left (a d \,x^{2}+a c +b \right )}{3 a^{2} d \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {\left (-\frac {2 d \left (a c +5 b \right ) \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}+\frac {\left (a^{2} c^{2}+a b c -3 b^{2}\right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{a \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {3 b^{2} \left (a c +b \right ) \left (-\frac {\left (a \,d^{2} x^{2}+a c d \right ) x}{\left (a c +b \right ) b d \sqrt {\left (x^{2}+\frac {a c +b}{a d}\right ) \left (a \,d^{2} x^{2}+a c d \right )}}+\frac {\left (\frac {1}{a c +b}+\frac {a c}{\left (a c +b \right ) b}\right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {2 a d \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{b \left (a c +b \right ) \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}\right )}{a}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{3 a^{2} d \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(831\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{2}}{\left (\frac {a c + a d x^{2} + b}{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 [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 {x^2}{{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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