3.2276 \(\int (d+e x)^{3/2} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=166 \[ \frac{2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^5}-\frac{4 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/
2))/(7*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(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 = 0.0733657, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {698} \[ \frac{2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^5}-\frac{4 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac{4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/
2))/(7*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(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 (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{e^4}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\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 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\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 = 0.149767, size = 174, normalized size = 1.05 \[ \frac{2 (d+e x)^{5/2} \left (143 e^2 \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-26 c e \left (3 b \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right )-11 a e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(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) + 143*e^2*(
63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 26*c*e*(-11*a*e*(8*d^2 - 20*d*e*
x + 35*e^2*x^2) + 3*b*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(45045*e^5)

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Maple [A]  time = 0.046, size = 194, normalized size = 1.2 \begin{align*}{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}+16380\,bc{e}^{4}{x}^{3}-5040\,{c}^{2}d{e}^{3}{x}^{3}+20020\,ac{e}^{4}{x}^{2}+10010\,{b}^{2}{e}^{4}{x}^{2}-10920\,bcd{e}^{3}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+25740\,ab{e}^{4}x-11440\,acd{e}^{3}x-5720\,{b}^{2}d{e}^{3}x+6240\,bc{d}^{2}{e}^{2}x-1920\,{c}^{2}{d}^{3}ex+18018\,{a}^{2}{e}^{4}-10296\,abd{e}^{3}+4576\,ac{d}^{2}{e}^{2}+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+a)^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+10010*a*c*e^4*x^2+5005*b^2*e^4*x^2
-5460*b*c*d*e^3*x^2+1680*c^2*d^2*e^2*x^2+12870*a*b*e^4*x-5720*a*c*d*e^3*x-2860*b^2*d*e^3*x+3120*b*c*d^2*e^2*x-
960*c^2*d^3*e*x+9009*a^2*e^4-5148*a*b*d*e^3+2288*a*c*d^2*e^2+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 = 0.978018, size = 238, normalized size = 1.43 \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 +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} 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+a)^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 + 2*a*c)*e^2)*(e*x + d)^(9/2) - 12870*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^
(7/2) + 9009*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*(e*x + d)^(5/2))/e^5

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Fricas [B]  time = 2.28634, size = 724, normalized size = 4.36 \begin{align*} \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e - 5148 \, a b d^{3} e^{3} + 9009 \, a^{2} d^{2} e^{4} + 1144 \,{\left (b^{2} + 2 \, a c\right )} 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 \,{\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{4} - 10 \,{\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 1287 \, a b e^{6} - 715 \,{\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 6864 \, a b d e^{5} + 3003 \, a^{2} e^{6} + 143 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{5} e - 312 \, b c d^{4} e^{2} - 1287 \, a b d^{2} e^{4} - 9009 \, a^{2} d e^{5} + 286 \,{\left (b^{2} + 2 \, a c\right )} 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+a)^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 - 5148*a*b*d^3*e^3 + 9009*a^2*d^2*e^4 + 1144*(b^2 + 2
*a*c)*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 + 2*a*c)*e^6
)*x^4 - 10*(12*c^2*d^3*e^3 - 39*b*c*d^2*e^4 - 1287*a*b*e^6 - 715*(b^2 + 2*a*c)*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2
- 156*b*c*d^3*e^3 + 6864*a*b*d*e^5 + 3003*a^2*e^6 + 143*(b^2 + 2*a*c)*d^2*e^4)*x^2 - 2*(96*c^2*d^5*e - 312*b*c
*d^4*e^2 - 1287*a*b*d^2*e^4 - 9009*a^2*d*e^5 + 286*(b^2 + 2*a*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 21.058, size = 654, normalized size = 3.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 4*a*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a*b*(d**2*(d + e*x)**(3/2)
/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*c*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 + 4*a*c*(-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 + 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 = 1.16917, size = 786, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d*e^(-1) + 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) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)
^(3/2)*d^2)*a*c*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) + 15015*(x*e + d)^(3/2)*a^2*d + 858*(15*(x
*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*e^(-1) + 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) + 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)*a*c*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) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)
^(3/2)*d)*a^2)*e^(-1)