Optimal. Leaf size=356 \[ \frac {a^{5/4} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B-63 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {6 a^{5/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {6 a A e^3 x \sqrt {a+c x^2}}{5 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c} \]
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Rubi [A] time = 0.41, antiderivative size = 356, 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 = {833, 842, 840, 1198, 220, 1196} \[ \frac {a^{5/4} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B-63 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {6 a^{5/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {6 a A e^3 x \sqrt {a+c x^2}}{5 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 833
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} (A+B x)}{\sqrt {a+c x^2}} \, dx &=\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 \int \frac {(e x)^{3/2} \left (-\frac {5}{2} a B e+\frac {7}{2} A c e x\right )}{\sqrt {a+c x^2}} \, dx}{7 c}\\ &=\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {e x} \left (-\frac {21}{4} a A c e^2-\frac {25}{4} a B c e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{35 c^2}\\ &=-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {8 \int \frac {\frac {25}{8} a^2 B c e^3-\frac {63}{8} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{105 c^3}\\ &=-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {25}{8} a^2 B c e^3-\frac {63}{8} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{105 c^3 \sqrt {e x}}\\ &=-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {\left (16 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {25}{8} a^2 B c e^3-\frac {63}{8} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 c^3 \sqrt {e x}}\\ &=-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {\left (2 a^{3/2} \left (25 \sqrt {a} B-63 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 )}{105 c^2 \sqrt {e x}}+\frac {\left (6 a^{3/2} 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 )}{5 c^{3/2} \sqrt {e x}}\\ &=-\frac {10 a B e^2 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{3/2} \sqrt {a+c x^2}}{5 c}+\frac {2 B (e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {6 a A e^3 x \sqrt {a+c x^2}}{5 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 a^{5/4} 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 )}{5 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{5/4} \left (25 \sqrt {a} B-63 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 )}{105 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 133, normalized size = 0.37 \[ \frac {2 e^2 \sqrt {e x} \left (25 a^2 B \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )-\left (a+c x^2\right ) (25 a B-3 c x (7 A+5 B x))-21 a A c x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{105 c^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{2} x^{3} + A e^{2} x^{2}\right )} \sqrt {e x}}{\sqrt {c x^{2} + a}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 336, normalized size = 0.94 \[ \frac {\sqrt {e x}\, \left (30 B \,c^{3} x^{5}+42 A \,c^{3} x^{4}-20 B a \,c^{2} x^{3}+42 A a \,c^{2} x^{2}-126 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A \,a^{2} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+63 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A \,a^{2} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-50 B \,a^{2} c x +25 \sqrt {-a c}\, \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{2}}{105 \sqrt {c \,x^{2}+a}\, c^{3} x} \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{5/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.01, size = 94, normalized size = 0.26 \[ \frac {A e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {B e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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