Optimal. Leaf size=369 \[ \frac {8 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3003 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {16 a^{13/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {8 a^2 \sqrt {e x} \sqrt {a+c x^2} (195 A+77 B x)}{3003 e}+\frac {20 a \sqrt {e x} \left (a+c x^2\right )^{3/2} (117 A+77 B x)}{9009 e}+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 A+11 B x)}{143 e} \]
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Rubi [A] time = 0.41, antiderivative size = 369, 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 = {815, 842, 840, 1198, 220, 1196} \[ \frac {8 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3003 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^2 \sqrt {e x} \sqrt {a+c x^2} (195 A+77 B x)}{3003 e}-\frac {16 a^{13/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {20 a \sqrt {e x} \left (a+c x^2\right )^{3/2} (117 A+77 B x)}{9009 e}+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 A+11 B x)}{143 e} \]
Antiderivative was successfully verified.
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Rule 220
Rule 815
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{\sqrt {e x}} \, dx &=\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {20 \int \frac {\left (\frac {13}{2} a A c e^2+\frac {11}{2} a B c e^2 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{143 c e^2}\\ &=\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {80 \int \frac {\left (\frac {117}{4} a^2 A c^2 e^4+\frac {77}{4} a^2 B c^2 e^4 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{3003 c^2 e^4}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {64 \int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{9009 c^3 e^6}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {\left (64 \sqrt {x}\right ) \int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{9009 c^3 e^6 \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {\left (128 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{9009 c^3 e^6 \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}-\frac {\left (16 a^{7/2} B \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 )}{39 \sqrt {c} \sqrt {e x}}+\frac {\left (16 a^3 \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3003 \sqrt {c} \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}-\frac {16 a^{13/4} B \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 )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{11/4} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) \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 )}{3003 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 85, normalized size = 0.23 \[ \frac {2 a^2 x \sqrt {a+c x^2} \left (3 A \, _2F_1\left (-\frac {5}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )+B x \, _2F_1\left (-\frac {5}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{3 \sqrt {e x} \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e x}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 362, normalized size = 0.98 \[ \frac {\frac {2 B \,c^{4} x^{8}}{13}+\frac {2 A \,c^{4} x^{7}}{11}+\frac {74 B a \,c^{3} x^{6}}{117}+\frac {62 A a \,c^{3} x^{5}}{77}+\frac {118 B \,a^{2} c^{2} x^{4}}{117}+\frac {122 A \,a^{2} c^{2} x^{3}}{77}+\frac {62 B \,a^{3} c \,x^{2}}{117}+\frac {74 A \,a^{3} c x}{77}+\frac {16 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{4} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{39}-\frac {8 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{4} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{39}+\frac {40 \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A \,a^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{77}}{\sqrt {c \,x^{2}+a}\, \sqrt {e x}\, c} \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{\sqrt {e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 18.20, size = 301, normalized size = 0.82 \[ \frac {A a^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {A a^{\frac {3}{2}} c x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {A \sqrt {a} c^{2} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} + \frac {B a^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {3}{2}} c x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\sqrt {e} \Gamma \left (\frac {11}{4}\right )} + \frac {B \sqrt {a} c^{2} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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