Optimal. Leaf size=147 \[ \frac {2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac {4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac {4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac {4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac {4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 \sqrt {d+e x}}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{3/2}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{7/2}}{e^4}+\frac {c^2 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 124, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 164, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (1155 b^2 d^2 e^2-1386 b^2 d e^2 (d+e x)+495 b^2 e^2 (d+e x)^2-2310 b c d^3 e+4158 b c d^2 e (d+e x)-2970 b c d e (d+e x)^2+770 b c e (d+e x)^3+1155 c^2 d^4-2772 c^2 d^3 (d+e x)+2970 c^2 d^2 (d+e x)^2-1540 c^2 d (d+e x)^3+315 c^2 (d+e x)^4\right )}{3465 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 175, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e + 264 \, b^{2} d^{3} e^{2} + 35 \, {\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, b^{2} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 33 \, b^{2} d e^{4}\right )} x^{2} - 4 \, {\left (16 \, c^{2} d^{4} e - 44 \, b c d^{3} e^{2} + 33 \, b^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 375, normalized size = 2.55 \begin {gather*} \frac {2}{3465} \, {\left (231 \, {\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^{2} d e^{\left (-2\right )} + 198 \, {\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 )} b c d e^{\left (-3\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{2} d e^{\left (-4\right )} + 99 \, {\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 )} b^{2} e^{\left (-2\right )} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b c e^{\left (-3\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 141, normalized size = 0.96 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} x^{4} e^{4}+770 b c \,e^{4} x^{3}-280 c^{2} d \,e^{3} x^{3}+495 b^{2} e^{4} x^{2}-660 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}-396 b^{2} d \,e^{3} x +528 b c \,d^{2} e^{2} x -192 c^{2} d^{3} e x +264 b^{2} d^{2} e^{2}-352 b c \,d^{3} e +128 c^{2} d^{4}\right )}{3465 e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 139, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} - 770 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{5\,e^5}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.07, size = 173, normalized size = 1.18 \begin {gather*} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________