Optimal. Leaf size=318 \[ -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x^{3/2} \left (b+c x^2\right )}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.27, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2063, 2048,
2057, 335, 311, 226, 1210} \begin {gather*} -\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {x^{5/2} (b B-3 A c)}{b^2 \sqrt {b x^2+c x^4}}-\frac {x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2048
Rule 2057
Rule 2063
Rubi steps
\begin {align*} \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {b B}{2}+\frac {3 A c}{2}\right )\right ) \int \frac {x^{7/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{b}\\ &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{2 b^2}\\ &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{2 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {c} \sqrt {b x^2+c x^4}}+\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {c} \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x^{3/2} \left (b+c x^2\right )}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 77, normalized size = 0.24 \begin {gather*} \frac {2 \sqrt {x} \left (-3 A b+(b B-3 A c) x^2 \sqrt {1+\frac {c x^2}{b}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {c x^2}{b}\right )\right )}{3 b^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 392, normalized size = 1.23
method | result | size |
default | \(\frac {x^{\frac {5}{2}} \left (c \,x^{2}+b \right ) \left (6 A \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, b c -3 A \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, b c -2 B \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, b^{2}+B \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, b^{2}-6 A \,c^{2} x^{2}+2 b B \,x^{2} c -4 A b c \right )}{2 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} c \,b^{2}}\) | \(392\) |
risch | \(-\frac {2 A \left (c \,x^{2}+b \right ) \sqrt {x}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {\left (\frac {A \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c \,x^{3}+b x}}-b \left (A c -B b \right ) \left (\frac {x^{2}}{b \sqrt {\left (x^{2}+\frac {b}{c}\right ) c x}}-\frac {\sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 b c \sqrt {c \,x^{3}+b x}}\right )\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(412\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 115, normalized size = 0.36 \begin {gather*} \frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{4} + {\left (B b^{2} - 3 \, A b c\right )} x^{2}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - \sqrt {c x^{4} + b x^{2}} {\left (2 \, A b c - {\left (B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt {x}}{b^{2} c^{2} x^{4} + b^{3} c x^{2}} \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^{\frac {3}{2}} \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 [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^{3/2}\,\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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