Optimal. Leaf size=109 \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 0.50 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (c d (7 d+5 e x)-2 a e^2\right )}{35 c^2 d^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 135, normalized size = 1.24 \[ \frac {2 \, {\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} + {\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} + {\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{35 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{\sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 0.63 \[ -\frac {2 \left (c d x +a e \right ) \left (-5 c d e x +2 a \,e^{2}-7 c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {3}{2}} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 98, normalized size = 0.90 \[ \frac {2 \, {\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} + {\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} + {\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt {c d x + a e}}{35 \, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 128, normalized size = 1.17 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x^2\,\left (14\,c^3\,d^4+16\,a\,c^2\,d^2\,e^2\right )}{35\,c^2\,d^2}-\frac {4\,a^3\,e^4-14\,a^2\,c\,d^2\,e^2}{35\,c^2\,d^2}+\frac {2\,c\,d\,e\,x^3}{7}+\frac {2\,a\,e\,x\,\left (14\,c\,d^2+a\,e^2\right )}{35\,c\,d}\right )}{\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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