3.356 $$\int \sqrt{d+e x} (b x+c x^2)^3 \, dx$$

Optimal. Leaf size=248 $\frac{6 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{9 e^7}+\frac{6 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}-\frac{6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac{2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}$

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))/(3*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e
^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7)

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Rubi [A]  time = 0.105513, antiderivative size = 248, normalized size of antiderivative = 1., 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} $\frac{6 c (d+e x)^{11/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{9 e^7}+\frac{6 d (d+e x)^{7/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}-\frac{6 d^2 (d+e x)^{5/2} (c d-b e)^2 (2 c d-b e)}{5 e^7}+\frac{2 d^3 (d+e x)^{3/2} (c d-b e)^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))/(3*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e
^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \sqrt{d+e x} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac{d^3 (c d-b e)^3 \sqrt{d+e x}}{e^6}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{7/2}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{11/2}}{e^6}+\frac{c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac{2 d^3 (c d-b e)^3 (d+e x)^{3/2}}{3 e^7}-\frac{6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac{6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac{6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end{align*}

Mathematica [A]  time = 0.15012, size = 206, normalized size = 0.83 $\frac{2 (d+e x)^{3/2} \left (12285 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-5005 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+19305 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-10395 c^2 (d+e x)^5 (2 c d-b e)-27027 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+15015 d^3 (c d-b e)^3+3003 c^3 (d+e x)^6\right )}{45045 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(15015*d^3*(c*d - b*e)^3 - 27027*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 19305*d*(c*d -
b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 5005*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(
d + e*x)^3 + 12285*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 10395*c^2*(2*c*d - b*e)*(d + e*x)^5 + 300
3*c^3*(d + e*x)^6))/(45045*e^7)

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Maple [A]  time = 0.048, size = 286, normalized size = 1.2 \begin{align*} -{\frac{-6006\,{c}^{3}{x}^{6}{e}^{6}-20790\,b{c}^{2}{e}^{6}{x}^{5}+5544\,{c}^{3}d{e}^{5}{x}^{5}-24570\,{b}^{2}c{e}^{6}{x}^{4}+18900\,b{c}^{2}d{e}^{5}{x}^{4}-5040\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-10010\,{b}^{3}{e}^{6}{x}^{3}+21840\,{b}^{2}cd{e}^{5}{x}^{3}-16800\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+4480\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+8580\,{b}^{3}d{e}^{5}{x}^{2}-18720\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+14400\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6864\,{b}^{3}{d}^{2}{e}^{4}x+14976\,{b}^{2}c{d}^{3}{e}^{3}x-11520\,b{c}^{2}{d}^{4}{e}^{2}x+3072\,{c}^{3}{d}^{5}ex+4576\,{b}^{3}{d}^{3}{e}^{3}-9984\,{b}^{2}c{d}^{4}{e}^{2}+7680\,b{c}^{2}{d}^{5}e-2048\,{c}^{3}{d}^{6}}{45045\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(c*x^2+b*x)^3,x)

[Out]

-2/45045*(e*x+d)^(3/2)*(-3003*c^3*e^6*x^6-10395*b*c^2*e^6*x^5+2772*c^3*d*e^5*x^5-12285*b^2*c*e^6*x^4+9450*b*c^
2*d*e^5*x^4-2520*c^3*d^2*e^4*x^4-5005*b^3*e^6*x^3+10920*b^2*c*d*e^5*x^3-8400*b*c^2*d^2*e^4*x^3+2240*c^3*d^3*e^
3*x^3+4290*b^3*d*e^5*x^2-9360*b^2*c*d^2*e^4*x^2+7200*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2-3432*b^3*d^2*e^4*x
+7488*b^2*c*d^3*e^3*x-5760*b*c^2*d^4*e^2*x+1536*c^3*d^5*e*x+2288*b^3*d^3*e^3-4992*b^2*c*d^4*e^2+3840*b*c^2*d^5
*e-1024*c^3*d^6)/e^7

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Maxima [A]  time = 1.13471, size = 366, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 10395 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 19305 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 27027 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^3 - 10395*(2*c^3*d - b*c^2*e)*(e*x + d)^(13/2) + 12285*(5*c^3*d^2 - 5*b*c^2*d
*e + b^2*c*e^2)*(e*x + d)^(11/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^(9/
2) + 19305*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(7/2) - 27027*(2*c^3*d^5 - 5*b
*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(5/2) + 15015*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2
- b^3*d^3*e^3)*(e*x + d)^(3/2))/e^7

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Fricas [A]  time = 2.23182, size = 720, normalized size = 2.9 \begin{align*} \frac{2 \,{\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e + 4992 \, b^{2} c d^{5} e^{2} - 2288 \, b^{3} d^{4} e^{3} + 231 \,{\left (c^{3} d e^{6} + 45 \, b c^{2} e^{7}\right )} x^{6} - 63 \,{\left (4 \, c^{3} d^{2} e^{5} - 15 \, b c^{2} d e^{6} - 195 \, b^{2} c e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \, b^{2} c d e^{6} + 143 \, b^{3} e^{7}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \, b^{2} c d^{2} e^{5} - 143 \, b^{3} d e^{6}\right )} x^{3} + 6 \,{\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} + 312 \, b^{2} c d^{3} e^{4} - 143 \, b^{3} d^{2} e^{5}\right )} x^{2} - 8 \,{\left (64 \, c^{3} d^{6} e - 240 \, b c^{2} d^{5} e^{2} + 312 \, b^{2} c d^{4} e^{3} - 143 \, b^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

2/45045*(3003*c^3*e^7*x^7 + 1024*c^3*d^7 - 3840*b*c^2*d^6*e + 4992*b^2*c*d^5*e^2 - 2288*b^3*d^4*e^3 + 231*(c^3
*d*e^6 + 45*b*c^2*e^7)*x^6 - 63*(4*c^3*d^2*e^5 - 15*b*c^2*d*e^6 - 195*b^2*c*e^7)*x^5 + 35*(8*c^3*d^3*e^4 - 30*
b*c^2*d^2*e^5 + 39*b^2*c*d*e^6 + 143*b^3*e^7)*x^4 - 5*(64*c^3*d^4*e^3 - 240*b*c^2*d^3*e^4 + 312*b^2*c*d^2*e^5
- 143*b^3*d*e^6)*x^3 + 6*(64*c^3*d^5*e^2 - 240*b*c^2*d^4*e^3 + 312*b^2*c*d^3*e^4 - 143*b^3*d^2*e^5)*x^2 - 8*(6
4*c^3*d^6*e - 240*b*c^2*d^5*e^2 + 312*b^2*c*d^4*e^3 - 143*b^3*d^3*e^4)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 6.26446, size = 326, normalized size = 1.31 \begin{align*} \frac{2 \left (\frac{c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 b c^{2} e - 6 c^{3} d\right )}{13 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (- 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(c*x**2+b*x)**3,x)

[Out]

2*(c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(13/2)*(3*b*c**2*e - 6*c**3*d)/(13*e**6) + (d + e*x)**(11/2)*
(3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(b**3*e**3 - 12*b**2*c*d*e**2 + 30
*b*c**2*d**2*e - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(-3*b**3*d*e**3 + 18*b**2*c*d**2*e**2 - 30*b*c**2*d
**3*e + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(3*b**3*d**2*e**3 - 12*b**2*c*d**3*e**2 + 15*b*c**2*d**4*e -
6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(-b**3*d**3*e**3 + 3*b**2*c*d**4*e**2 - 3*b*c**2*d**5*e + c**3*d**6)
/(3*e**6))/e

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Giac [A]  time = 1.33636, size = 410, normalized size = 1.65 \begin{align*} \frac{2}{45045} \,{\left (143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b^{3} e^{\left (-3\right )} + 39 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b^{2} c e^{\left (-4\right )} + 15 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} b c^{2} e^{\left (-5\right )} +{\left (3003 \,{\left (x e + d\right )}^{\frac{15}{2}} - 20790 \,{\left (x e + d\right )}^{\frac{13}{2}} d + 61425 \,{\left (x e + d\right )}^{\frac{11}{2}} d^{2} - 100100 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{3} + 96525 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{4} - 54054 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{5} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{6}\right )} c^{3} e^{\left (-6\right )}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/45045*(143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*
b^3*e^(-3) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/
2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*c*e^(-4) + 15*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(
x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b*c^2*e^
(-5) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)
*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^3*e^(-6))*e^(-1)