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

Optimal. Leaf size=286 \[ \frac {6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac {6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 286, 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} \begin {gather*} \frac {6 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^7}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^7}+\frac {6 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac {6 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.37, size = 396, normalized size = 1.38 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (39 c e^2 \left (33 a^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 a b e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^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 )+143 e^3 \left (105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )-3 c^2 e \left (5 b \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )-13 a e \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 )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 27
72*d*e^5*x^5 + 3003*e^6*x^6) + 143*e^3*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) + 9*a*b^2*e*(8*d^2 - 12*d*e*
x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)) + 39*c*e^2*(33*a^2*e^2*(8*d^2 - 12*d
*e*x + 15*e^2*x^2) + 22*a*b*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + b^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)) - 3*c^2*e*(-13*a*e*(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) + 5*b*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^
4 - 693*e^5*x^5))))/(45045*e^7)

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IntegrateAlgebraic [B]  time = 0.23, size = 592, normalized size = 2.07 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+27027 a^2 b e^5 (d+e x)-45045 a^2 b d e^5+45045 a^2 c d^2 e^4-54054 a^2 c d e^4 (d+e x)+19305 a^2 c e^4 (d+e x)^2+45045 a b^2 d^2 e^4-54054 a b^2 d e^4 (d+e x)+19305 a b^2 e^4 (d+e x)^2-90090 a b c d^3 e^3+162162 a b c d^2 e^3 (d+e x)-115830 a b c d e^3 (d+e x)^2+30030 a b c e^3 (d+e x)^3+45045 a c^2 d^4 e^2-108108 a c^2 d^3 e^2 (d+e x)+115830 a c^2 d^2 e^2 (d+e x)^2-60060 a c^2 d e^2 (d+e x)^3+12285 a c^2 e^2 (d+e x)^4-15015 b^3 d^3 e^3+27027 b^3 d^2 e^3 (d+e x)-19305 b^3 d e^3 (d+e x)^2+5005 b^3 e^3 (d+e x)^3+45045 b^2 c d^4 e^2-108108 b^2 c d^3 e^2 (d+e x)+115830 b^2 c d^2 e^2 (d+e x)^2-60060 b^2 c d e^2 (d+e x)^3+12285 b^2 c e^2 (d+e x)^4-45045 b c^2 d^5 e+135135 b c^2 d^4 e (d+e x)-193050 b c^2 d^3 e (d+e x)^2+150150 b c^2 d^2 e (d+e x)^3-61425 b c^2 d e (d+e x)^4+10395 b c^2 e (d+e x)^5+15015 c^3 d^6-54054 c^3 d^5 (d+e x)+96525 c^3 d^4 (d+e x)^2-100100 c^3 d^3 (d+e x)^3+61425 c^3 d^2 (d+e x)^4-20790 c^3 d (d+e x)^5+3003 c^3 (d+e x)^6\right )}{45045 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(15015*c^3*d^6 - 45045*b*c^2*d^5*e + 45045*b^2*c*d^4*e^2 + 45045*a*c^2*d^4*e^2 - 15015*b^3*
d^3*e^3 - 90090*a*b*c*d^3*e^3 + 45045*a*b^2*d^2*e^4 + 45045*a^2*c*d^2*e^4 - 45045*a^2*b*d*e^5 + 15015*a^3*e^6
- 54054*c^3*d^5*(d + e*x) + 135135*b*c^2*d^4*e*(d + e*x) - 108108*b^2*c*d^3*e^2*(d + e*x) - 108108*a*c^2*d^3*e
^2*(d + e*x) + 27027*b^3*d^2*e^3*(d + e*x) + 162162*a*b*c*d^2*e^3*(d + e*x) - 54054*a*b^2*d*e^4*(d + e*x) - 54
054*a^2*c*d*e^4*(d + e*x) + 27027*a^2*b*e^5*(d + e*x) + 96525*c^3*d^4*(d + e*x)^2 - 193050*b*c^2*d^3*e*(d + e*
x)^2 + 115830*b^2*c*d^2*e^2*(d + e*x)^2 + 115830*a*c^2*d^2*e^2*(d + e*x)^2 - 19305*b^3*d*e^3*(d + e*x)^2 - 115
830*a*b*c*d*e^3*(d + e*x)^2 + 19305*a*b^2*e^4*(d + e*x)^2 + 19305*a^2*c*e^4*(d + e*x)^2 - 100100*c^3*d^3*(d +
e*x)^3 + 150150*b*c^2*d^2*e*(d + e*x)^3 - 60060*b^2*c*d*e^2*(d + e*x)^3 - 60060*a*c^2*d*e^2*(d + e*x)^3 + 5005
*b^3*e^3*(d + e*x)^3 + 30030*a*b*c*e^3*(d + e*x)^3 + 61425*c^3*d^2*(d + e*x)^4 - 61425*b*c^2*d*e*(d + e*x)^4 +
12285*b^2*c*e^2*(d + e*x)^4 + 12285*a*c^2*e^2*(d + e*x)^4 - 20790*c^3*d*(d + e*x)^5 + 10395*b*c^2*e*(d + e*x)
^5 + 3003*c^3*(d + e*x)^6))/(45045*e^7)

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fricas [A]  time = 0.41, size = 512, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 1024 \, c^{3} d^{7} - 3840 \, b c^{2} d^{6} e - 18018 \, a^{2} b d^{2} e^{5} + 15015 \, a^{3} d e^{6} + 4992 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} - 2288 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} + 10296 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 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 \, {\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} - 30 \, b c^{2} d^{2} e^{5} + 39 \, {\left (b^{2} c + a c^{2}\right )} d e^{6} + 143 \, {\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} - 240 \, b c^{2} d^{3} e^{4} + 312 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} - 143 \, {\left (b^{3} + 6 \, a b c\right )} d e^{6} - 3861 \, {\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{5} e^{2} - 480 \, b c^{2} d^{4} e^{3} + 9009 \, a^{2} b e^{7} + 624 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 286 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} + 1287 \, {\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{6} e - 1920 \, b c^{2} d^{5} e^{2} - 9009 \, a^{2} b d e^{6} - 15015 \, a^{3} e^{7} + 2496 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 1144 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 5148 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3*(e*x+d)^(1/2),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 - 18018*a^2*b*d^2*e^5 + 15015*a^3*d*e^6 + 4992*(b^
2*c + a*c^2)*d^5*e^2 - 2288*(b^3 + 6*a*b*c)*d^4*e^3 + 10296*(a*b^2 + a^2*c)*d^3*e^4 + 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 + a*c^2)*e^7)*x^5 + 35*(8*c^3*d^3*e^4 - 30*b*c^2*
d^2*e^5 + 39*(b^2*c + a*c^2)*d*e^6 + 143*(b^3 + 6*a*b*c)*e^7)*x^4 - 5*(64*c^3*d^4*e^3 - 240*b*c^2*d^3*e^4 + 31
2*(b^2*c + a*c^2)*d^2*e^5 - 143*(b^3 + 6*a*b*c)*d*e^6 - 3861*(a*b^2 + a^2*c)*e^7)*x^3 + 3*(128*c^3*d^5*e^2 - 4
80*b*c^2*d^4*e^3 + 9009*a^2*b*e^7 + 624*(b^2*c + a*c^2)*d^3*e^4 - 286*(b^3 + 6*a*b*c)*d^2*e^5 + 1287*(a*b^2 +
a^2*c)*d*e^6)*x^2 - (512*c^3*d^6*e - 1920*b*c^2*d^5*e^2 - 9009*a^2*b*d*e^6 - 15015*a^3*e^7 + 2496*(b^2*c + a*c
^2)*d^4*e^3 - 1144*(b^3 + 6*a*b*c)*d^3*e^4 + 5148*(a*b^2 + a^2*c)*d^2*e^5)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.26, size = 1248, normalized size = 4.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d*e^(-2) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*a^2*c*d*e^(-2) + 1287*(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^3*d*e^(-3) + 7722*(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*b*c*d*e^(-3) + 429*(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^2*c*d*e^(-4) + 429*(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)*a*c^2*d*e^(-4) + 195*(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)*b*c^2*d*e^(-5) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e +
d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e
+ d)*d^6)*c^3*d*e^(-6) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b*e^(-1) +
3861*(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*b^2*e^(-2)
+ 3861*(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^2*c*e^(-2)
+ 143*(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*s
qrt(x*e + d)*d^4)*b^3*e^(-3) + 858*(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)*a*b*c*e^(-3) + 195*(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)*b^2*c
*e^(-4) + 195*(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)*a*c^2*e^(-4) + 45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)
^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3
/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b*c^2*e^(-5) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*
e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*
(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*e^(-6) + 45045*sqrt(x*e + d)*a^3*d + 15015*((x*e + d)^(3/2)
- 3*sqrt(x*e + d)*d)*a^3)*e^(-1)

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maple [A]  time = 0.05, size = 495, normalized size = 1.73 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 c^{3} x^{6} e^{6}+10395 b \,c^{2} e^{6} x^{5}-2772 c^{3} d \,e^{5} x^{5}+12285 a \,c^{2} e^{6} x^{4}+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}+30030 a b c \,e^{6} x^{3}-10920 a \,c^{2} d \,e^{5} x^{3}+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}+19305 a^{2} c \,e^{6} x^{2}+19305 a \,b^{2} e^{6} x^{2}-25740 a b c d \,e^{5} x^{2}+9360 a \,c^{2} d^{2} e^{4} x^{2}-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}+27027 a^{2} b \,e^{6} x -15444 a^{2} c d \,e^{5} x -15444 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -7488 a \,c^{2} d^{3} e^{3} x +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 +15015 a^{3} e^{6}-18018 a^{2} b d \,e^{5}+10296 a^{2} c \,d^{2} e^{4}+10296 a \,b^{2} d^{2} e^{4}-13728 a b c \,d^{3} e^{3}+4992 a \,c^{2} d^{4} e^{2}-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}\right )}{45045 e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3*(e*x+d)^(1/2),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*a*c^2*e^6*x^4+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+30030*a*b*c*e^6*x^3-10920*a*c^2*d*e^5*x^3+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+19305*a^2*c*e^6*x^2+19305*a*b^2*e^6*x^2-2574
0*a*b*c*d*e^5*x^2+9360*a*c^2*d^2*e^4*x^2-4290*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+27027*a^2*b*e^6*x-15444*a^2*c*d*e^5*x-15444*a*b^2*d*e^5*x+20592*a*b*c*d^2*e^4*x-7488*a*c^2*d^
3*e^3*x+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+15015*a^3*e^6-18018*a^2*
b*d*e^5+10296*a^2*c*d^2*e^4+10296*a*b^2*d^2*e^4-13728*a*b*c*d^3*e^3+4992*a*c^2*d^4*e^2-2288*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 = 0.93, size = 407, normalized size = 1.42 \begin {gather*} \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 + {\left (b^{2} c + a c^{2}\right )} 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 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} 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 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3*(e*x+d)^(1/2),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 + a*c^2)*e^2)*(e*x + d)^(11/2) - 5005*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b
^3 + 6*a*b*c)*e^3)*(e*x + d)^(9/2) + 19305*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*
a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(7/2) - 27027*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c
+ a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(5/2) + 15015*(c^3*d^6 - 3*b*c
^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d
^2*e^4)*(e*x + d)^(3/2))/e^7

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mupad [B]  time = 0.85, size = 297, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{11\,e^7}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{3\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{9\,e^7}+\frac {6\,\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 )}^2}{5\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(7/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 36*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 -
60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(7*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d +
e*x)^(13/2))/(13*e^7) + ((d + e*x)^(11/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(11*e^7) +
(2*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(3*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e^2 + 10*c^2*d^2
+ 6*a*c*e^2 - 10*b*c*d*e))/(9*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^7)

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sympy [A]  time = 8.93, size = 539, normalized size = 1.88 \begin {gather*} \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 a c^{2} e^{2} + 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 (6 a b c e^{3} - 12 a c^{2} d e^{2} + 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 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{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}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{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}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{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}\right )}{3 e^{6}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3*(e*x+d)**(1/2),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*a*c**2*e**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)*(6*a*b*c*e**3 - 12
*a*c**2*d*e**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*a**2*c*e**4 + 3*a*b**2*e**4 - 18*a*b*c*d*e**3 + 18*a*c**2*d**2*e**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*a**2*b*e**5 - 6*a**2*c*d*e**4 - 6*a*b**2*d*e**
4 + 18*a*b*c*d**2*e**3 - 12*a*c**2*d**3*e**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)*(a**3*e**6 - 3*a**2*b*d*e**5 + 3*a**2*c*d**2*e**4 + 3*a*b**2*d**2*e**4 -
6*a*b*c*d**3*e**3 + 3*a*c**2*d**4*e**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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