Optimal. Leaf size=357 \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (9 \sqrt {a} B-5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{9/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}+\frac {3 B \sqrt {c} x \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (9 \sqrt {a} B-5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{9/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}+\frac {3 B \sqrt {c} x \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
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)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {5}{2} a A c e^2-\frac {3}{2} a B c e^2 x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}+\frac {2 \int \frac {\frac {9}{4} a^2 B c e^3-\frac {5}{4} a A c^2 e^3 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a^3 c e^4}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}-\frac {4 \int \frac {\frac {5}{8} a^2 A c^2 e^4-\frac {9}{8} a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a^4 c e^6}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}-\frac {\left (4 \sqrt {x}\right ) \int \frac {\frac {5}{8} a^2 A c^2 e^4-\frac {9}{8} a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a^4 c e^6 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}-\frac {\left (8 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {5}{8} a^2 A c^2 e^4-\frac {9}{8} a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^4 c e^6 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}-\frac {\left (3 B \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^2 \sqrt {e x}}+\frac {\left (\left (9 \sqrt {a} B-5 A \sqrt {c}\right ) \sqrt {c} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^2 e^2 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac {3 B \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x}}+\frac {3 B \sqrt {c} x \sqrt {a+c x^2}}{a^2 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 B \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^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (9 \sqrt {a} B-5 A \sqrt {c}\right ) \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{9/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 107, normalized size = 0.30 \[ \frac {x \left (3 \left (-3 B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )+A+B x\right )-5 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {c x^2}{a}\right )\right )}{3 a (e x)^{5/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{2} e^{3} x^{7} + 2 \, a c e^{3} x^{5} + a^{2} e^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 307, normalized size = 0.86 \[ -\frac {18 B c \,x^{3}+10 A c \,x^{2}-18 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a x \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+9 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+5 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {-a c}\, A x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+12 B a x +4 A a}{6 \sqrt {c \,x^{2}+a}\, \sqrt {e x}\, a^{2} e^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{{\left (e\,x\right )}^{5/2}\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 65.58, size = 100, normalized size = 0.28 \[ \frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________