Optimal. Leaf size=341 \[ \frac {4 a^{3/4} \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (9 \sqrt {a} B+5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {24 a^{5/4} B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 \sqrt {a+c x^2} (9 a B-5 A c x)}{15 e^2 \sqrt {e x}}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A-3 B x)}{15 e (e x)^{3/2}}+\frac {24 a B \sqrt {c} x \sqrt {a+c x^2}}{5 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rubi [A] time = 0.32, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {813, 842, 840, 1198, 220, 1196} \[ \frac {4 a^{3/4} \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (9 \sqrt {a} B+5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {24 a^{5/4} B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 \sqrt {a+c x^2} (9 a B-5 A c x)}{15 e^2 \sqrt {e x}}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A-3 B x)}{15 e (e x)^{3/2}}+\frac {24 a B \sqrt {c} x \sqrt {a+c x^2}}{5 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 813
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{5/2}} \, dx &=-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}-\frac {2 \int \frac {(-3 a B e-5 A c e x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx}{5 e^2}\\ &=-\frac {4 (9 a B-5 A c x) \sqrt {a+c x^2}}{15 e^2 \sqrt {e x}}-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}+\frac {4 \int \frac {5 a A c e^2+9 a B c e^2 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 e^4}\\ &=-\frac {4 (9 a B-5 A c x) \sqrt {a+c x^2}}{15 e^2 \sqrt {e x}}-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}+\frac {\left (4 \sqrt {x}\right ) \int \frac {5 a A c e^2+9 a B c e^2 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 e^4 \sqrt {e x}}\\ &=-\frac {4 (9 a B-5 A c x) \sqrt {a+c x^2}}{15 e^2 \sqrt {e x}}-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}+\frac {\left (8 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {5 a A c e^2+9 a B c e^2 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 e^4 \sqrt {e x}}\\ &=-\frac {4 (9 a B-5 A c x) \sqrt {a+c x^2}}{15 e^2 \sqrt {e x}}-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}-\frac {\left (24 a^{3/2} 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 )}{5 e^2 \sqrt {e x}}+\frac {\left (8 a \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 )}{15 e^2 \sqrt {e x}}\\ &=\frac {24 a B \sqrt {c} x \sqrt {a+c x^2}}{5 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 (9 a B-5 A c x) \sqrt {a+c x^2}}{15 e^2 \sqrt {e x}}-\frac {2 (5 A-3 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{3/2}}-\frac {24 a^{5/4} 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 )}{5 e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 a^{3/4} \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 )}{15 e^2 \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 83, normalized size = 0.24 \[ -\frac {2 a x \sqrt {a+c x^2} \left (A \, _2F_1\left (-\frac {3}{2},-\frac {3}{4};\frac {1}{4};-\frac {c x^2}{a}\right )+3 B x \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-\frac {c x^2}{a}\right )\right )}{3 (e x)^{5/2} \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 325, normalized size = 0.95 \[ \frac {\frac {2 B \,c^{2} x^{5}}{5}+\frac {2 A \,c^{2} x^{4}}{3}-\frac {8 B a c \,x^{3}}{5}+\frac {24 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} x \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {12 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {4 \sqrt {-a c}\, \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A a x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3}-2 B \,a^{2} x -\frac {2 A \,a^{2}}{3}}{\sqrt {c \,x^{2}+a}\, \sqrt {e x}\, e^{2} x} \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.70, size = 206, normalized size = 0.60 \[ \frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {A \sqrt {a} c \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} c x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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