Optimal. Leaf size=125 \[ \frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {794, 648} \begin {gather*} \frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \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 &=\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}+\frac {1}{7} \left (7 f-\frac {5 d g}{e}-\frac {2 a e g}{c d}\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\\ &=\frac {2 \left (7 f-\frac {5 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 c d (d+e x)^{5/2}}+\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 0.43 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+5 g x)-2 a e g)}{35 c^2 d^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.31, size = 67, normalized size = 0.54 \begin {gather*} \frac {2 (a e+c d x) ((d+e x) (a e+c d x))^{3/2} (5 g (a e+c d x)-7 a e g+7 c d f)}{35 c^2 d^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 137, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (5 \, c^{3} d^{3} g x^{3} + 7 \, a^{2} c d e^{2} f - 2 \, a^{3} e^{3} g + {\left (7 \, c^{3} d^{3} f + 8 \, a c^{2} d^{2} e g\right )} x^{2} + {\left (14 \, a c^{2} d^{2} e f + a^{2} c d e^{2} g\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 67, normalized size = 0.54 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-5 c d g x +2 a e g -7 c d f \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 107, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d x + a e} f}{5 \, c d} + \frac {2 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} g}{35 \, c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 109, normalized size = 0.87 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^2\,\left (\frac {16\,a\,e\,g}{35}+\frac {2\,c\,d\,f}{5}\right )-\frac {4\,a^3\,e^3\,g-14\,a^2\,c\,d\,e^2\,f}{35\,c^2\,d^2}+\frac {2\,c\,d\,g\,x^3}{7}+\frac {2\,a\,e\,x\,\left (a\,e\,g+14\,c\,d\,f\right )}{35\,c\,d}\right )}{\sqrt {d+e\,x}} \end {gather*}
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 \begin {gather*} \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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