Optimal. Leaf size=388 \[ \frac {a^{7/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {14 a^{9/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c} \]
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Rubi [A] time = 0.48, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {a^{7/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c} \]
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)^{7/2} (A+B x)}{\sqrt {a+c x^2}} \, dx &=\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {2 \int \frac {(e x)^{5/2} \left (-\frac {7}{2} a B e+\frac {9}{2} A c e x\right )}{\sqrt {a+c x^2}} \, dx}{9 c}\\ &=\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {4 \int \frac {(e x)^{3/2} \left (-\frac {45}{4} a A c e^2-\frac {49}{4} a B c e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{63 c^2}\\ &=-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {8 \int \frac {\sqrt {e x} \left (\frac {147}{8} a^2 B c e^3-\frac {225}{8} a A c^2 e^3 x\right )}{\sqrt {a+c x^2}} \, dx}{315 c^3}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {16 \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{945 c^4}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (16 \sqrt {x}\right ) \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (32 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}-\frac {\left (14 a^{5/2} B e^4 \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 )}{15 c^{5/2} \sqrt {e x}}+\frac {\left (2 a^2 \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 c^{5/2} \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^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 )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{7/4} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^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 )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 142, normalized size = 0.37 \[ \frac {2 e^3 \sqrt {e x} \left (75 a^2 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )+49 a^2 B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )-\left (a+c x^2\right ) \left (a (75 A+49 B x)-5 c x^2 (9 A+7 B x)\right )\right )}{315 c^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt {e x}}{\sqrt {c x^{2} + a}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 344, normalized size = 0.89 \[ \frac {\sqrt {e x}\, \left (70 B \,c^{3} x^{6}+90 A \,c^{3} x^{5}-28 B a \,c^{2} x^{4}-60 A a \,c^{2} x^{3}-98 B \,a^{2} c \,x^{2}-150 A \,a^{2} c x +294 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-147 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+75 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-a c}\, A \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{3}}{315 \sqrt {c \,x^{2}+a}\, c^{3} x} \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{7/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 55.52, size = 94, normalized size = 0.24 \[ \frac {A e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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