Optimal. Leaf size=356 \[ \frac {8 b (b B+9 A c) x^{3/2} \left (b+c x^2\right )}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac {8 b^{5/4} (b B+9 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{5/4} (b B+9 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.31, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2063, 2046,
2057, 335, 311, 226, 1210} \begin {gather*} \frac {4 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}-\frac {8 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 A c+b B) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {4}{15} \sqrt {x} \sqrt {b x^2+c x^4} (9 A c+b B)+\frac {2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac {8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2046
Rule 2057
Rule 2063
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{9/2}} \, dx &=-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac {\left (2 \left (-\frac {b B}{2}-\frac {9 A c}{2}\right )\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{5/2}} \, dx}{b}\\ &=\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {1}{3} (2 (b B+9 A c)) \int \frac {\sqrt {b x^2+c x^4}}{\sqrt {x}} \, dx\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {1}{15} (4 b (b B+9 A c)) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (4 b (b B+9 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 \sqrt {b x^2+c x^4}}\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (8 b (b B+9 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 )}{15 \sqrt {b x^2+c x^4}}\\ &=\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac {\left (8 b^{3/2} (b B+9 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 )}{15 \sqrt {c} \sqrt {b x^2+c x^4}}-\frac {\left (8 b^{3/2} (b B+9 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 )}{15 \sqrt {c} \sqrt {b x^2+c x^4}}\\ &=\frac {8 b (b B+9 A c) x^{3/2} \left (b+c x^2\right )}{15 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {4}{15} (b B+9 A c) \sqrt {x} \sqrt {b x^2+c x^4}+\frac {2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac {8 b^{5/4} (b B+9 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 )}{15 c^{3/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{5/4} (b B+9 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 )}{15 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 = 10.05, size = 85, normalized size = 0.24 \begin {gather*} \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (-\frac {3 A \left (b+c x^2\right )^2}{b}+\frac {(b B+9 A c) x^2 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )}{\sqrt {1+\frac {c x^2}{b}}}\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 429, normalized size = 1.21
method | result | size |
risch | \(-\frac {2 \left (-5 B c \,x^{4}-9 A c \,x^{2}-11 b B \,x^{2}+45 A b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{45 x^{\frac {3}{2}}}+\frac {4 b \left (9 A c +B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{15 c \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(249\) |
default | \(\frac {2 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} \left (5 B \,c^{3} x^{6}+108 A \,b^{2} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-54 A \,b^{2} c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+12 B \,b^{3} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-6 B \,b^{3} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+9 A \,c^{3} x^{4}+16 B b \,c^{2} x^{4}-36 A b \,c^{2} x^{2}+11 B \,b^{2} c \,x^{2}-45 A \,b^{2} c \right )}{45 x^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{2} c}\) | \(429\) |
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 = 0.78, size = 94, normalized size = 0.26 \begin {gather*} -\frac {2 \, {\left (12 \, {\left (B b^{2} + 9 \, A b c\right )} \sqrt {c} x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - {\left (5 \, B c^{2} x^{4} - 45 \, A b c + {\left (11 \, B b c + 9 \, A c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{45 \, 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 {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{\frac {9}{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 {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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