3.2277 \(\int \sqrt{d+e x} (a+b x+c x^2)^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0724684, 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} \[ \frac{2 (d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac{4 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/
2))/(5*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(7/2))/(7*e^5) - (4*c*(2*c*d - b*e)*(d
+ e*x)^(9/2))/(9*e^5) + (2*c^2*(d + e*x)^(11/2))/(11*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 \sqrt{d+e x} \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 \sqrt{d+e x}}{e^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{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)^{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 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 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)^{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 = 0.151066, size = 172, normalized size = 1.04 \[ \frac{2 (d+e x)^{3/2} \left (33 e^2 \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-22 c e \left (b \left (-24 d^2 e x+16 d^3+30 d e^2 x^2-35 e^3 x^3\right )-3 a e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+c^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(c^2*(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) + 33*e^2*(35*a
^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 22*c*e*(-3*a*e*(8*d^2 - 12*d*e*x + 1
5*e^2*x^2) + b*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3))))/(3465*e^5)

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Maple [A]  time = 0.043, size = 194, normalized size = 1.2 \begin{align*}{\frac{630\,{c}^{2}{x}^{4}{e}^{4}+1540\,bc{e}^{4}{x}^{3}-560\,{c}^{2}d{e}^{3}{x}^{3}+1980\,ac{e}^{4}{x}^{2}+990\,{b}^{2}{e}^{4}{x}^{2}-1320\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+2772\,ab{e}^{4}x-1584\,acd{e}^{3}x-792\,{b}^{2}d{e}^{3}x+1056\,bc{d}^{2}{e}^{2}x-384\,{c}^{2}{d}^{3}ex+2310\,{a}^{2}{e}^{4}-1848\,abd{e}^{3}+1056\,ac{d}^{2}{e}^{2}+528\,{b}^{2}{d}^{2}{e}^{2}-704\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(3/2)*(315*c^2*e^4*x^4+770*b*c*e^4*x^3-280*c^2*d*e^3*x^3+990*a*c*e^4*x^2+495*b^2*e^4*x^2-660*b*
c*d*e^3*x^2+240*c^2*d^2*e^2*x^2+1386*a*b*e^4*x-792*a*c*d*e^3*x-396*b^2*d*e^3*x+528*b*c*d^2*e^2*x-192*c^2*d^3*e
*x+1155*a^2*e^4-924*a*b*d*e^3+528*a*c*d^2*e^2+264*b^2*d^2*e^2-352*b*c*d^3*e+128*c^2*d^4)/e^5

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Maxima [A]  time = 0.971869, size = 238, normalized size = 1.43 \begin{align*} \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 +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\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{5}{2}} + 1155 \,{\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{3}{2}}\right )}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*c^2 - 770*(2*c^2*d - b*c*e)*(e*x + d)^(9/2) + 495*(6*c^2*d^2 - 6*b*c*d*e + (b^2 +
2*a*c)*e^2)*(e*x + d)^(7/2) - 1386*(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)^(5/2)
+ 1155*(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)^(3/2))/e^5

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Fricas [A]  time = 2.31287, size = 554, normalized size = 3.34 \begin{align*} \frac{2 \,{\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e - 924 \, a b d^{2} e^{3} + 1155 \, a^{2} d e^{4} + 264 \,{\left (b^{2} + 2 \, a c\right )} 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 \,{\left (b^{2} + 2 \, a c\right )} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 462 \, a b e^{5} + 33 \,{\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{2} -{\left (64 \, c^{2} d^{4} e - 176 \, b c d^{3} e^{2} - 462 \, a b d e^{4} - 1155 \, a^{2} e^{5} + 132 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c^2*e^5*x^5 + 128*c^2*d^5 - 352*b*c*d^4*e - 924*a*b*d^2*e^3 + 1155*a^2*d*e^4 + 264*(b^2 + 2*a*c)*d
^3*e^2 + 35*(c^2*d*e^4 + 22*b*c*e^5)*x^4 - 5*(8*c^2*d^2*e^3 - 22*b*c*d*e^4 - 99*(b^2 + 2*a*c)*e^5)*x^3 + 3*(16
*c^2*d^3*e^2 - 44*b*c*d^2*e^3 + 462*a*b*e^5 + 33*(b^2 + 2*a*c)*d*e^4)*x^2 - (64*c^2*d^4*e - 176*b*c*d^3*e^2 -
462*a*b*d*e^4 - 1155*a^2*e^5 + 132*(b^2 + 2*a*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 4.05104, size = 230, normalized size = 1.39 \begin{align*} \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 (2 a c e^{2} + 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 a b e^{3} - 4 a c d e^{2} - 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 (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16171, size = 338, normalized size = 2.04 \begin{align*} \frac{2}{3465} \,{\left (462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b e^{\left (-1\right )} + 33 \,{\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} e^{\left (-2\right )} + 66 \,{\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 )} a c e^{\left (-2\right )} + 22 \,{\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 e^{\left (-3\right )} +{\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} e^{\left (-4\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*e^(-1) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)
*d + 35*(x*e + d)^(3/2)*d^2)*b^2*e^(-2) + 66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d
^2)*a*c*e^(-2) + 22*(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*e^(-3) + (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*e^(-4) + 1155*(x*e + d)^(3/2)*a^2)*e^(-1)