Optimal. Leaf size=68 \[ -\frac {2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac {2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac {2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) \sqrt {d+e x}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{3/2}}{e^2}+\frac {c (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 d (c d-b e) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 50, normalized size = 0.74 \[ \frac {2 (d+e x)^{3/2} \left (7 b e (3 e x-2 d)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 71, normalized size = 1.04 \[ \frac {2 \, {\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 3 \, {\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} - {\left (4 \, c d^{2} e - 7 \, b d e^{2}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 165, normalized size = 2.43 \[ \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d e^{\left (-2\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b e^{\left (-1\right )} + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c e^{\left (-2\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 47, normalized size = 0.69 \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-15 c \,e^{2} x^{2}-21 b \,e^{2} x +12 c d e x +14 b d e -8 c \,d^{2}\right )}{105 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 54, normalized size = 0.79 \[ \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c - 21 \, {\left (2 \, c d - b e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (c d^{2} - b d e\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 52, normalized size = 0.76 \[ \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,c\,{\left (d+e\,x\right )}^2+35\,c\,d^2+21\,b\,e\,\left (d+e\,x\right )-42\,c\,d\,\left (d+e\,x\right )-35\,b\,d\,e\right )}{105\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.67, size = 66, normalized size = 0.97 \[ \frac {2 \left (\frac {c \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________