Optimal. Leaf size=363 \[ -\frac {c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {6 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {6 A c^{3/2} x \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {6 A c^{3/2} x \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 835
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {A+B x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 \int \frac {-\frac {5}{2} a B e+\frac {3}{2} A c e x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx}{5 a e^2}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {4 \int \frac {-\frac {9}{4} a A c e^2-\frac {5}{4} a B c e^2 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{15 a^2 e^4}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {8 \int \frac {\frac {5}{8} a^2 B c e^3+\frac {9}{8} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 a^3 e^6}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {5}{8} a^2 B c e^3+\frac {9}{8} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 a^3 e^6 \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {\left (16 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {5}{8} a^2 B c e^3+\frac {9}{8} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a^3 e^6 \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {\left (2 \left (5 \sqrt {a} B+9 A \sqrt {c}\right ) c \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a^{3/2} e^3 \sqrt {e x}}+\frac {\left (6 A c^{3/2} \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 a^{3/2} e^3 \sqrt {e x}}\\ &=-\frac {2 A \sqrt {a+c x^2}}{5 a e (e x)^{5/2}}-\frac {2 B \sqrt {a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac {6 A c \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {6 A c^{3/2} x \sqrt {a+c x^2}}{5 a^2 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 A c^{5/4} \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 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (5 \sqrt {a} B+9 A \sqrt {c}\right ) c^{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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 83, normalized size = 0.23 \[ -\frac {2 x \sqrt {\frac {c x^2}{a}+1} \left (3 A \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};-\frac {c x^2}{a}\right )+5 B x \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c e^{4} x^{6} + a e^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 331, normalized size = 0.91 \[ -\frac {-18 A \,c^{2} x^{4}+18 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A a c \,x^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-9 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A a c \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+10 B a c \,x^{3}-12 A a c \,x^{2}+5 \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+10 B \,a^{2} x +6 A \,a^{2}}{15 \sqrt {c \,x^{2}+a}\, \sqrt {e x}\, a^{2} e^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{{\left (e\,x\right )}^{7/2}\,\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 29.24, size = 104, normalized size = 0.29 \[ \frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________