Optimal. Leaf size=296 \[ \frac {e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B+A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {3 B e^2 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {819, 842, 840, 1198, 220, 1196} \[ \frac {e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B+A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {3 B e^2 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 819
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {\frac {1}{2} a A e^2+\frac {3}{2} a B e^2 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\sqrt {x} \int \frac {\frac {1}{2} a A e^2+\frac {3}{2} a B e^2 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a c \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\left (2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} a A e^2+\frac {3}{2} a B e^2 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a c \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}-\frac {\left (3 \sqrt {a} B e^2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{c^{3/2} \sqrt {e x}}+\frac {\left (2 \left (\frac {3}{2} a B e^2+\frac {1}{2} \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} c^{3/2} \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{c \sqrt {a+c x^2}}+\frac {3 B e^2 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} B e^2 \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 )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) e^2 \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 \sqrt [4]{a} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 102, normalized size = 0.34 \[ \frac {e \sqrt {e x} \left (A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )+B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )-A-B x\right )}{c \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e x^{2} + A e x\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 290, normalized size = 0.98 \[ \frac {\sqrt {e x}\, \left (-2 B c \,x^{2}-2 A c x +6 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {-a c}\, A \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right ) e}{2 \sqrt {c \,x^{2}+a}\, c^{2} x} \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 28.55, size = 94, normalized size = 0.32 \[ \frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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