Optimal. Leaf size=118 \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {794, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 c e^3}\\ &=-\frac {2 (7 c e f-3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 78, normalized size = 0.66 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} (c (2 d g+7 e f+5 e g x)-2 b e g)}{35 c^2 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.59, size = 74, normalized size = 0.63 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{5/2} (-2 b e g+5 c g (d+e x)-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 229, normalized size = 1.94 \begin {gather*} -\frac {2 \, {\left (5 \, c^{3} e^{3} g x^{3} + {\left (7 \, c^{3} e^{3} f - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + 2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g - {\left (14 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f - {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{35 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 79, normalized size = 0.67 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-5 c e g x +2 b e g -2 c d g -7 c e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {3}{2}} c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.75, size = 197, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (c^{2} e^{2} x^{2} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{5 \, c e} - \frac {2 \, {\left (5 \, c^{3} e^{3} x^{3} + 2 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{35 \, c^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.81, size = 133, normalized size = 1.13 \begin {gather*} -\frac {\left (x^2\,\left (\frac {16\,b\,e\,g}{35}-\frac {16\,c\,d\,g}{35}+\frac {2\,c\,e\,f}{5}\right )+\frac {2\,c\,e\,g\,x^3}{7}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\left (2\,c\,d\,g-2\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e^2}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\left (b\,e\,g-c\,d\,g+14\,c\,e\,f\right )}{35\,c\,e}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e 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.
[In]
[Out]
________________________________________________________________________________________