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]

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

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Rubi [A]  time = 0.07, antiderivative size = 166, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.14, size = 172, normalized size = 1.04 \[ \frac {2 (d+e x)^{3/2} \left (33 e^2 \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-22 c e \left (b \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 a e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]

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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fricas [A]  time = 0.93, size = 237, normalized size = 1.43 \[ \frac {2 \, {\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e - 924 \, a b d^{2} e^{3} + 1155 \, a^{2} d e^{4} + 264 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + 35 \, {\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 462 \, a b e^{5} + 33 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{4} e - 176 \, b c d^{3} e^{2} - 462 \, a b d e^{4} - 1155 \, a^{2} e^{5} + 132 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \]

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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giac [B]  time = 0.18, size = 578, normalized size = 3.48 \[ \frac {2}{3465} \, {\left (2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b d e^{\left (-1\right )} + 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 )} + 462 \, {\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 )} a c 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 )} + 462 \, {\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 )} a b e^{\left (-1\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 )} + 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 )} a c 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 )} + 3465 \, \sqrt {x e + d} a^{2} d + 1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2}\right )} e^{\left (-1\right )} \]

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*(2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*d*e^(-1) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*
d + 15*sqrt(x*e + d)*d^2)*b^2*d*e^(-2) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*a*c*d*e^(-2) + 198*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)
*b*c*d*e^(-3) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)
*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*d*e^(-4) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*
d^2)*a*b*e^(-1) + 99*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3
)*b^2*e^(-2) + 198*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*
a*c*e^(-2) + 22*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^
3 + 315*sqrt(x*e + d)*d^4)*b*c*e^(-3) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d
^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c^2*e^(-4) + 3465*sqrt(x*e +
d)*a^2*d + 1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2)*e^(-1)

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maple [A]  time = 0.06, size = 194, normalized size = 1.17 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} x^{4} e^{4}+770 b c \,e^{4} x^{3}-280 c^{2} d \,e^{3} x^{3}+990 a c \,e^{4} x^{2}+495 b^{2} e^{4} x^{2}-660 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+1386 a b \,e^{4} x -792 a c d \,e^{3} x -396 b^{2} d \,e^{3} x +528 b c \,d^{2} e^{2} x -192 c^{2} d^{3} e x +1155 a^{2} e^{4}-924 a b d \,e^{3}+528 a c \,d^{2} e^{2}+264 b^{2} d^{2} e^{2}-352 b c \,d^{3} e +128 c^{2} d^{4}\right )}{3465 e^{5}} \]

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 = 0.88, size = 176, normalized size = 1.06 \[ \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} - 770 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \]

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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mupad [B]  time = 0.05, size = 148, normalized size = 0.89 \[ \frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{7\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{5\,e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(11/2))/(11*e^5) + ((d + e*x)^(7/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(7*e^5
) + (2*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(3*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(9/2))/(9*e^5) + (4
*(b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e))/(5*e^5)

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sympy [A]  time = 5.14, size = 230, normalized size = 1.39 \[ \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \]

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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