### 3.347 $$\int (d+e x)^{3/2} (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=147 $\frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}$

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*c^2*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.0606813, antiderivative size = 147, 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} $\frac{2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac{2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac{4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(9*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*c^2*(d + e*x)^(13/2))/(13*e^5)

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 (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^{3/2}}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{5/2}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{9/2}}{e^4}+\frac{c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 (d+e x)^{5/2}}{5 e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0775826, size = 125, normalized size = 0.85 $\frac{2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (40 d^2 e x-16 d^3-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^
2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5
)

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Maple [A]  time = 0.05, size = 141, normalized size = 1. \begin{align*}{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}+16380\,bc{e}^{4}{x}^{3}-5040\,{c}^{2}d{e}^{3}{x}^{3}+10010\,{b}^{2}{e}^{4}{x}^{2}-10920\,bcd{e}^{3}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-5720\,{b}^{2}d{e}^{3}x+6240\,bc{d}^{2}{e}^{2}x-1920\,{c}^{2}{d}^{3}ex+2288\,{b}^{2}{d}^{2}{e}^{2}-2496\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4+8190*b*c*e^4*x^3-2520*c^2*d*e^3*x^3+5005*b^2*e^4*x^2-5460*b*c*d*e^3*x^
2+1680*c^2*d^2*e^2*x^2-2860*b^2*d*e^3*x+3120*b*c*d^2*e^2*x-960*c^2*d^3*e*x+1144*b^2*d^2*e^2-1248*b*c*d^3*e+384
*c^2*d^4)/e^5

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Maxima [A]  time = 1.218, size = 188, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 8190 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 8190*(2*c^2*d - b*c*e)*(e*x + d)^(11/2) + 5005*(6*c^2*d^2 - 6*b*c*d*e + b
^2*e^2)*(e*x + d)^(9/2) - 12870*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(7/2) + 9009*(c^2*d^4 - 2*b*c*
d^3*e + b^2*d^2*e^2)*(e*x + d)^(5/2))/e^5

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Fricas [A]  time = 1.89437, size = 493, normalized size = 3.35 \begin{align*} \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e + 1144 \, b^{2} d^{4} e^{2} + 630 \,{\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, b^{2} e^{6}\right )} x^{4} - 10 \,{\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 715 \, b^{2} d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 143 \, b^{2} d^{2} e^{4}\right )} x^{2} - 4 \,{\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 143 \, b^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*c^2*e^6*x^6 + 384*c^2*d^6 - 1248*b*c*d^5*e + 1144*b^2*d^4*e^2 + 630*(7*c^2*d*e^5 + 13*b*c*e^6)*x
^5 + 35*(3*c^2*d^2*e^4 + 312*b*c*d*e^5 + 143*b^2*e^6)*x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 715*b^2*d*e^
5)*x^3 + 3*(48*c^2*d^4*e^2 - 156*b*c*d^3*e^3 + 143*b^2*d^2*e^4)*x^2 - 4*(48*c^2*d^5*e - 156*b*c*d^4*e^2 + 143*
b^2*d^3*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [B]  time = 18.7187, size = 413, normalized size = 2.81 \begin{align*} \frac{2 b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{4 b c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{4 b c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} + \frac{2 c^{2} d \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{5}} + \frac{2 c^{2} \left (- \frac{d^{5} \left (d + e x\right )^{\frac{3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac{5}{2}} - \frac{10 d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{10 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9} - \frac{5 d \left (d + e x\right )^{\frac{11}{2}}}{11} + \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*b**2*(-d**3*(d + e*x
)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*b*c*d*(-d**3*(d
+ e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 4*b*c*(d**4
*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*
x)**(11/2)/11)/e**4 + 2*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(
5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/
13)/e**5

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Giac [B]  time = 1.42503, size = 506, normalized size = 3.44 \begin{align*} \frac{2}{45045} \,{\left (429 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{2} d e^{\left (-2\right )} + 286 \,{\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 c d e^{\left (-3\right )} + 13 \,{\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 )} c^{2} d e^{\left (-4\right )} + 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^{2} e^{\left (-2\right )} + 26 \,{\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 c e^{\left (-3\right )} + 5 \,{\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 )} c^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d*e^(-2) + 286*(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*c*d*e^(-3) + 13*(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)*c^2*d*e^(-4) + 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^2*e^(-2) + 26*(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*c*e^(-3) + 5*(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)*c^2*e^(-4))*e^(-1)