Optimal. Leaf size=327 \[ \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {823, 835, 842, 840, 1198, 220, 1196} \[ \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 823
Rule 835
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {3}{2} a A c e^2-\frac {1}{2} a B c e^2 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {2 \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (2 \sqrt {x}\right ) \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (4 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} e \sqrt {e x}}-\frac {\left (3 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 )}{a^{3/2} e \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 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^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+3 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 )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 100, normalized size = 0.31 \[ \frac {x \left (-3 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )+B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )+A+B x\right )}{a (e x)^{3/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{2} e^{2} x^{6} + 2 \, a c e^{2} x^{4} + a^{2} e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 304, normalized size = 0.93 \[ -\frac {6 A \,c^{2} x^{2}-6 \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, A a c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-2 B a c x +4 A a c -\sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {2}\, B a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {c \,x^{2}+a}\, \sqrt {e x}\, a^{2} c e} \]
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 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 29.21, size = 97, normalized size = 0.30 \[ \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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