Optimal. Leaf size=388 \[ -\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{7/4} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.33, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {847, 856, 854,
1212, 226, 1210} \begin {gather*} \frac {a^{7/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {14 a^{9/4} B e^4 \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 )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 847
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} (A+B x)}{\sqrt {a+c x^2}} \, dx &=\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {2 \int \frac {(e x)^{5/2} \left (-\frac {7}{2} a B e+\frac {9}{2} A c e x\right )}{\sqrt {a+c x^2}} \, dx}{9 c}\\ &=\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {4 \int \frac {(e x)^{3/2} \left (-\frac {45}{4} a A c e^2-\frac {49}{4} a B c e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{63 c^2}\\ &=-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {8 \int \frac {\sqrt {e x} \left (\frac {147}{8} a^2 B c e^3-\frac {225}{8} a A c^2 e^3 x\right )}{\sqrt {a+c x^2}} \, dx}{315 c^3}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {16 \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{945 c^4}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (16 \sqrt {x}\right ) \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (32 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}-\frac {\left (14 a^{5/2} B e^4 \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 )}{15 c^{5/2} \sqrt {e x}}+\frac {\left (2 a^2 \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 c^{5/2} \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^4 \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 )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{7/4} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \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 )}{105 c^{11/4} \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 = 10.07, size = 142, normalized size = 0.37 \begin {gather*} \frac {2 e^3 \sqrt {e x} \left (-\left (\left (a+c x^2\right ) \left (-5 c x^2 (9 A+7 B x)+a (75 A+49 B x)\right )\right )+75 a^2 A \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 )+49 a^2 B x \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{315 c^2 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 344, normalized size = 0.89
method | result | size |
default | \(\frac {e^{3} \sqrt {e x}\, \left (70 B \,c^{3} x^{6}+75 A \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+90 A \,c^{3} x^{5}+294 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-147 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-28 a B \,c^{2} x^{4}-60 a A \,c^{2} x^{3}-98 a^{2} B c \,x^{2}-150 a^{2} A c x \right )}{315 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(344\) |
risch | \(-\frac {2 \left (-35 B c \,x^{3}-45 A c \,x^{2}+49 B a x +75 A a \right ) x \sqrt {c \,x^{2}+a}\, e^{4}}{315 c^{2} \sqrt {e x}}+\frac {a^{2} \left (\frac {49 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {25 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{4} \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(359\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{3} x^{3} \sqrt {c e \,x^{3}+a e x}}{9 c}+\frac {2 A \,e^{3} x^{2} \sqrt {c e \,x^{3}+a e x}}{7 c}-\frac {14 B \,e^{3} a x \sqrt {c e \,x^{3}+a e x}}{45 c^{2}}-\frac {10 A \,e^{3} a \sqrt {c e \,x^{3}+a e x}}{21 c^{2}}+\frac {5 A \,e^{4} a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {7 B \,e^{4} a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{15 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(416\) |
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 = 0.66, size = 99, normalized size = 0.26 \begin {gather*} \frac {2 \, {\left (75 \, A a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 147 \, B a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (35 \, B c^{2} x^{3} + 45 \, A c^{2} x^{2} - 49 \, B a c x - 75 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {7}{2}}\right )}}{315 \, c^{3}} \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 = 44.81, size = 94, normalized size = 0.24 \begin {gather*} \frac {A e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {15}{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 {{\left (e\,x\right )}^{7/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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