Optimal. Leaf size=125 \[ \frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt {d+e x}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \]
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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 )^{3/2}}{5 c d e \sqrt {d+e x}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt {d+e x}}+\frac {1}{5} \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx\\ &=\frac {2 \left (5 f-\frac {3 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 )^{3/2}}{15 c d (d+e x)^{3/2}}+\frac {2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 0.43 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+3 g x)-2 a e g)}{15 c^2 d^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 169, normalized size = 1.35 \begin {gather*} \frac {2 \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}} \left (-2 a^2 e^4 g-a c d^2 e^2 g+5 a c d e^3 f+a c d e^2 g (d+e x)+3 c^2 d^4 g-5 c^2 d^3 e f-6 c^2 d^3 g (d+e x)+5 c^2 d^2 e f (d+e x)+3 c^2 d^2 g (d+e x)^2\right )}{15 c^2 d^2 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 102, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 2 \, a^{2} e^{2} g + {\left (5 \, c^{2} d^{2} f + a c d e 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}}{15 \, {\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}}{\sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.54 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-3 c d g x +2 a e g -5 c d f \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \sqrt {e x +d}\, c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 65, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f}{3 \, c d} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} g}{15 \, c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.13, size = 93, normalized size = 0.74 \begin {gather*} \frac {\left (\frac {2\,g\,x^2}{5}-\frac {4\,a^2\,e^2\,g-10\,a\,c\,d\,e\,f}{15\,c^2\,d^2}+\frac {x\,\left (10\,f\,c^2\,d^2+2\,a\,e\,g\,c\,d\right )}{15\,c^2\,d^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\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 {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}{\sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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