Optimal. Leaf size=298 \[ \frac {e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}-\frac {A e x \sqrt {a+c x^2}}{a \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 298, 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 = {821, 842, 840, 1198, 220, 1196} \[ \frac {e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}-\frac {A e x \sqrt {a+c x^2}}{a \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 821
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}+\frac {\int \frac {\frac {a B e}{2}-\frac {1}{2} A c e x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}+\frac {\sqrt {x} \int \frac {\frac {a B e}{2}-\frac {1}{2} A c e x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a c \sqrt {e x}}\\ &=-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}+\frac {\left (2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {a B e}{2}-\frac {1}{2} A c e x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a c \sqrt {e x}}\\ &=-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) e \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} c \sqrt {e x}}+\frac {\left (A e \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 )}{\sqrt {a} \sqrt {c} \sqrt {e x}}\\ &=-\frac {\sqrt {e x} (a B-A c x)}{a c \sqrt {a+c x^2}}-\frac {A e x \sqrt {a+c x^2}}{a \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A e \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^{3/4} c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) e \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^{3/4} c^{5/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 113, normalized size = 0.38 \[ \frac {\sqrt {e x} \left (-A c x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )+3 a B \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )-3 a B+3 A c x\right )}{3 a c \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 297, normalized size = 1.00 \[ \frac {\sqrt {e x}\, \left (2 A \,c^{2} x^{2}-2 \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-2 B a c x +\sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, B a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right )}{2 \sqrt {c \,x^{2}+a}\, a \,c^{2} x} \]
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 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {e\,x}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.19, size = 94, normalized size = 0.32 \[ \frac {A \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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