Optimal. Leaf size=438 \[ \frac {4 a^{13/4} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15015 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{13/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^3 A e^3 x \sqrt {a+c x^2}}{65 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 a^2 e^2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{3003 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c} \]
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Rubi [A] time = 0.58, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {833, 815, 842, 840, 1198, 220, 1196} \[ \frac {4 a^2 e^2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15015 c^2}+\frac {4 a^{13/4} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15015 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^3 A e^3 x \sqrt {a+c x^2}}{65 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {8 a^{13/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a e^2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{3003 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 815
Rule 833
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {2 \int (e x)^{3/2} \left (-\frac {5}{2} a B e+\frac {15}{2} A c e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{15 c}\\ &=\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {4 \int \sqrt {e x} \left (-\frac {45}{4} a A c e^2-\frac {65}{4} a B c e^2 x\right ) \left (a+c x^2\right )^{3/2} \, dx}{195 c^2}\\ &=-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {8 \int \frac {\left (\frac {65}{8} a^2 B c e^3-\frac {495}{8} a A c^2 e^3 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{2145 c^3}\\ &=\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {32 \int \frac {\left (\frac {585}{16} a^3 B c^2 e^5-\frac {3465}{16} a^2 A c^3 e^5 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{45045 c^4 e^2}\\ &=\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {128 \int \frac {\frac {2925}{32} a^4 B c^3 e^7-\frac {10395}{32} a^3 A c^4 e^7 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{675675 c^5 e^4}\\ &=\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {\left (128 \sqrt {x}\right ) \int \frac {\frac {2925}{32} a^4 B c^3 e^7-\frac {10395}{32} a^3 A c^4 e^7 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{675675 c^5 e^4 \sqrt {e x}}\\ &=\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {\left (256 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {2925}{32} a^4 B c^3 e^7-\frac {10395}{32} a^3 A c^4 e^7 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{675675 c^5 e^4 \sqrt {e x}}\\ &=\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {\left (8 a^{7/2} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) e^3 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15015 c^2 \sqrt {e x}}+\frac {\left (8 a^{7/2} A e^3 \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 )}{65 c^{3/2} \sqrt {e x}}\\ &=-\frac {8 a^3 A e^3 x \sqrt {a+c x^2}}{65 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {8 a^{13/4} A e^3 \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 )}{65 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 a^{13/4} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) e^3 \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 )}{15015 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 137, normalized size = 0.31 \[ \frac {2 e^2 \sqrt {e x} \sqrt {a+c x^2} \left (65 a^3 B \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )-165 a^2 A c x \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )-\left (a+c x^2\right )^2 \sqrt {\frac {c x^2}{a}+1} (65 a B-11 c x (15 A+13 B x))\right )}{2145 c^2 \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B c e^{2} x^{5} + A c e^{2} x^{4} + B a e^{2} x^{3} + A a e^{2} x^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 384, normalized size = 0.88 \[ \frac {2 \sqrt {e x}\, \left (1001 B \,c^{5} x^{9}+1155 A \,c^{5} x^{8}+2548 B a \,c^{4} x^{7}+3080 A a \,c^{4} x^{6}+1703 B \,a^{2} c^{3} x^{5}+2233 A \,a^{2} c^{3} x^{4}-104 B \,a^{3} c^{2} x^{3}+308 A \,a^{3} c^{2} x^{2}-924 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A \,a^{4} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+462 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A \,a^{4} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-260 B \,a^{4} c x +130 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {-a c}\, B \,a^{4} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{2}}{15015 \sqrt {c \,x^{2}+a}\, c^{3} x} \]
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 \[ \int {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (e\,x\right )}^{5/2}\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 68.10, size = 199, normalized size = 0.45 \[ \frac {A a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {A \sqrt {a} c e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} + \frac {B a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} c e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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