Optimal. Leaf size=297 \[ \frac {2 a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B+5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 a^{5/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+3 B x)}{15 e}+\frac {4 a B x \sqrt {a+c x^2}}{5 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 297, 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 = {815, 842, 840, 1198, 220, 1196} \[ \frac {2 a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B+5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 a^{5/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (5 A+3 B x)}{15 e}+\frac {4 a B x \sqrt {a+c x^2}}{5 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 815
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx &=\frac {2 \sqrt {e x} (5 A+3 B x) \sqrt {a+c x^2}}{15 e}+\frac {4 \int \frac {\frac {5}{2} a A c e^2+\frac {3}{2} a B c e^2 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 c e^2}\\ &=\frac {2 \sqrt {e x} (5 A+3 B x) \sqrt {a+c x^2}}{15 e}+\frac {\left (4 \sqrt {x}\right ) \int \frac {\frac {5}{2} a A c e^2+\frac {3}{2} a B c e^2 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 c e^2 \sqrt {e x}}\\ &=\frac {2 \sqrt {e x} (5 A+3 B x) \sqrt {a+c x^2}}{15 e}+\frac {\left (8 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {5}{2} a A c e^2+\frac {3}{2} a B c e^2 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 c e^2 \sqrt {e x}}\\ &=\frac {2 \sqrt {e x} (5 A+3 B x) \sqrt {a+c x^2}}{15 e}-\frac {\left (4 a^{3/2} B \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 )}{5 \sqrt {c} \sqrt {e x}}+\frac {\left (4 a \left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {c} \sqrt {e x}}\\ &=\frac {2 \sqrt {e x} (5 A+3 B x) \sqrt {a+c x^2}}{15 e}+\frac {4 a B x \sqrt {a+c x^2}}{5 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a^{5/4} B \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 )}{5 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a^{3/4} \left (3 \sqrt {a} B+5 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 )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 82, normalized size = 0.28 \[ \frac {2 x \sqrt {a+c x^2} \left (3 A \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )+B x \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{3 \sqrt {e x} \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{e x}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 312, normalized size = 1.05 \[ \frac {\frac {2 B \,c^{2} x^{4}}{5}+\frac {2 A \,c^{2} x^{3}}{3}+\frac {2 B a c \,x^{2}}{5}+\frac {2 A a c x}{3}+\frac {4 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {2 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {-a c}\, A a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3}}{\sqrt {c \,x^{2}+a}\, \sqrt {e x}\, c} \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{\sqrt {e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.56, size = 97, normalized size = 0.33 \[ \frac {A \sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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