Optimal. Leaf size=360 \[ -\frac {a^{3/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (63 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {21 a^{5/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}-\frac {21 a B e^4 x \sqrt {a+c x^2}}{5 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2} \]
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Rubi [A] time = 0.40, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {819, 833, 842, 840, 1198, 220, 1196} \[ -\frac {a^{3/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (63 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {21 a^{5/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}-\frac {21 a B e^4 x \sqrt {a+c x^2}}{5 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 819
Rule 833
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {(e x)^{3/2} \left (\frac {5}{2} a A e^2+\frac {7}{2} a B e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}+\frac {2 \int \frac {\sqrt {e x} \left (-\frac {21}{4} a^2 B e^3+\frac {25}{4} a A c e^3 x\right )}{\sqrt {a+c x^2}} \, dx}{5 a c^2}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}+\frac {4 \int \frac {-\frac {25}{8} a^2 A c e^4-\frac {63}{8} a^2 B c e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 a c^3}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}+\frac {\left (4 \sqrt {x}\right ) \int \frac {-\frac {25}{8} a^2 A c e^4-\frac {63}{8} a^2 B c e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 a c^3 \sqrt {e x}}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}+\frac {\left (8 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {25}{8} a^2 A c e^4-\frac {63}{8} a^2 B c e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a c^3 \sqrt {e x}}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}+\frac {\left (21 a^{3/2} 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 )}{5 c^{5/2} \sqrt {e x}}-\frac {\left (a \left (63 \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 )}{15 c^{5/2} \sqrt {e x}}\\ &=-\frac {e (e x)^{5/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 A e^3 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {7 B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{5 c^2}-\frac {21 a B e^4 x \sqrt {a+c x^2}}{5 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 a^{5/4} 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 )}{5 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {a^{3/4} \left (63 \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 )}{30 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 127, normalized size = 0.35 \[ \frac {e^3 \sqrt {e x} \left (-25 a A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )+25 a A-21 a B x \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )+21 a B x+10 A c x^2+6 B c x^3\right )}{15 c^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 318, normalized size = 0.88 \[ -\frac {\sqrt {e x}\, \left (-12 B \,c^{2} x^{4}-20 A \,c^{2} x^{3}-42 B a c \,x^{2}-50 A a c x +126 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-63 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+25 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {-a c}\, A a \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{3}}{30 \sqrt {c \,x^{2}+a}\, c^{3} x} \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{7/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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