Optimal. Leaf size=393 \[ -\frac {c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {21 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.50, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}} \]
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)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {7}{2} a A c e^2-\frac {5}{2} a B c e^2 x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}+\frac {2 \int \frac {\frac {25}{4} a^2 B c e^3-\frac {21}{4} a A c^2 e^3 x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx}{5 a^3 c e^4}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}-\frac {4 \int \frac {\frac {63}{8} a^2 A c^2 e^4+\frac {25}{8} a^2 B c^2 e^4 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{15 a^4 c e^6}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {8 \int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 a^5 c e^8}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {\left (8 \sqrt {x}\right ) \int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 a^5 c e^8 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {\left (16 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a^5 c e^8 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {\left (\left (25 \sqrt {a} B+63 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^{5/2} e^3 \sqrt {e x}}+\frac {\left (21 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^{5/2} e^3 \sqrt {e x}}\\ &=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 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^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (25 \sqrt {a} B+63 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 )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 107, normalized size = 0.27 \[ \frac {x \left (-21 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};-\frac {c x^2}{a}\right )-25 B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {c x^2}{a}\right )+15 (A+B x)\right )}{15 a (e x)^{7/2} \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{2} e^{4} x^{8} + 2 \, a c e^{4} x^{6} + a^{2} e^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 331, normalized size = 0.84 \[ -\frac {-126 A \,c^{2} x^{4}+126 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A a c \,x^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-63 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A a c \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+50 B a c \,x^{3}-84 A a c \,x^{2}+25 \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+20 B \,a^{2} x +12 A \,a^{2}}{30 \sqrt {c \,x^{2}+a}\, \sqrt {e x}\, a^{3} e^{3} x^{2}} \]
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 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,x}{{\left (e\,x\right )}^{7/2}\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 150.96, size = 104, normalized size = 0.26 \[ \frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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