Optimal. Leaf size=327 \[ \frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {837, 849, 856,
854, 1212, 226, 1210} \begin {gather*} \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 837
Rule 849
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps
\begin {align*} \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {3}{2} a A c e^2-\frac {1}{2} a B c e^2 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {2 \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (2 \sqrt {x}\right ) \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} e \sqrt {e x}}-\frac {\left (3 A \sqrt {c} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} e \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}\\ \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 = 100, normalized size = 0.31 \begin {gather*} \frac {x \left (A+B x-3 A \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )+B x \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )\right )}{a (e x)^{3/2} \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 304, normalized size = 0.93
method | result | size |
default | \(-\frac {3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c -6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c -B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a +6 A \,c^{2} x^{2}-2 a B c x +4 A a c}{2 \sqrt {c \,x^{2}+a}\, c e \sqrt {e x}\, a^{2}}\) | \(304\) |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right ) A}{e^{2} a^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {2 c e x \left (\frac {A x}{2 e^{2} a^{2}}-\frac {B}{2 e^{2} a c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) c e x}}+\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 a e c \sqrt {c e \,x^{3}+a e x}}+\frac {3 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a^{2} e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(386\) |
risch | \(-\frac {2 A \sqrt {c \,x^{2}+a}}{a^{2} e \sqrt {e x}}+\frac {\left (\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}-a \left (-\frac {2 x c e \left (-\frac {A x}{2 e a}+\frac {B}{2 e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) c e x}}-\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 c \sqrt {c e \,x^{3}+a e x}}-\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{a^{2} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(532\) |
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 = 0.52, size = 121, normalized size = 0.37 \begin {gather*} \frac {{\left ({\left (B a c x^{3} + B a^{2} x\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3 \, {\left (A c^{2} x^{3} + A a c x\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3 \, A c^{2} x^{2} - B a c x + 2 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{a^{2} c^{2} x^{3} + a^{3} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 12.12, size = 97, normalized size = 0.30 \begin {gather*} \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \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 {A+B\,x}{{\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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