Optimal. Leaf size=293 \[ \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}+\frac {2 A \sqrt {c} x \sqrt {a+c x^2}}{a e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rubi [A] time = 0.25, antiderivative size = 293, 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 = {835, 842, 840, 1198, 220, 1196} \[ \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}+\frac {2 A \sqrt {c} x \sqrt {a+c x^2}}{a e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 835
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {A+B x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx &=-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}-\frac {2 \int \frac {-\frac {1}{2} a B e-\frac {1}{2} A c e x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a e^2}\\ &=-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}-\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {1}{2} a B e-\frac {1}{2} A c e x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a e^2 \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}-\frac {\left (4 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} a B e-\frac {1}{2} A c e x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a e^2 \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}+\frac {\left (2 \left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} e \sqrt {e x}}-\frac {\left (2 A \sqrt {c} \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} e \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{a e \sqrt {e x}}+\frac {2 A \sqrt {c} x \sqrt {a+c x^2}}{a e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 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^{3/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+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 )}{a^{3/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 0.27 \[ \frac {2 x \sqrt {\frac {c x^2}{a}+1} \left (B x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )-A \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )\right )}{(e x)^{3/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c e^{2} x^{4} + a e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 296, normalized size = 1.01 \[ \frac {-2 A \,c^{2} x^{2}+2 \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-2 A a c +\sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, B a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {c \,x^{2}+a}\, \sqrt {e x}\, a c e} \]
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 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.21, size = 97, normalized size = 0.33 \[ \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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