3.2285 \(\int \frac {(a+b x+c x^2)^3}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=282 \[ \frac {2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]

[Out]

-2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(3/2)/e^7+6/5*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5
*b*d))*(e*x+d)^(5/2)/e^7-2/7*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(7/2)/e^7+2/3*c*(5
*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(9/2)/e^7-6/11*c^2*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^7+2/13*c^3*(e*x+d)
^(13/2)/e^7+2*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(1/2)/e^7

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Rubi [A]  time = 0.13, antiderivative size = 282, 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} \[ \frac {2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.55, size = 317, normalized size = 1.12 \[ \frac {2 \sqrt {d+e x} (a+x (b+c x))^3}{e}-\frac {4 (d+e x)^{3/2} \left (-286 c e^2 \left (21 a^2 e^2 (2 d-3 e x)-9 a b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^2 \left (32 d^3-48 d^2 e x+60 d e^2 x^2-70 e^3 x^3\right )\right )+429 b e^3 \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+13 c^2 e \left (44 a e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-10 c^3 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )}{15015 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(a + x*(b + c*x))^3)/e - (4*(d + e*x)^(3/2)*(-10*c^3*(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) + 429*b*e^3*(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)) - 286*c*e^2*(21*a^2*e^2*(2*d - 3*e*x) - 9*a*b*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) +
b^2*(32*d^3 - 48*d^2*e*x + 60*d*e^2*x^2 - 70*e^3*x^3)) + 13*c^2*e*(44*a*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^
2 + 35*e^3*x^3) + 5*b*(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))))/(15015*e^7)

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fricas [A]  time = 0.89, size = 408, normalized size = 1.45 \[ \frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e - 30030 \, a^{2} b d e^{5} + 15015 \, a^{3} e^{6} + 18304 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 6864 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 24024 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 315 \, {\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 429 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} - 2080 \, b c^{2} d^{3} e^{3} + 2288 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 858 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3003 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{5} e - 8320 \, b c^{2} d^{4} e^{2} - 15015 \, a^{2} b e^{6} + 9152 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3432 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 12012 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \]

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]

2/15015*(1155*c^3*e^6*x^6 + 5120*c^3*d^6 - 16640*b*c^2*d^5*e - 30030*a^2*b*d*e^5 + 15015*a^3*e^6 + 18304*(b^2*
c + a*c^2)*d^4*e^2 - 6864*(b^3 + 6*a*b*c)*d^3*e^3 + 24024*(a*b^2 + a^2*c)*d^2*e^4 - 315*(4*c^3*d*e^5 - 13*b*c^
2*e^6)*x^5 + 35*(40*c^3*d^2*e^4 - 130*b*c^2*d*e^5 + 143*(b^2*c + a*c^2)*e^6)*x^4 - 5*(320*c^3*d^3*e^3 - 1040*b
*c^2*d^2*e^4 + 1144*(b^2*c + a*c^2)*d*e^5 - 429*(b^3 + 6*a*b*c)*e^6)*x^3 + 3*(640*c^3*d^4*e^2 - 2080*b*c^2*d^3
*e^3 + 2288*(b^2*c + a*c^2)*d^2*e^4 - 858*(b^3 + 6*a*b*c)*d*e^5 + 3003*(a*b^2 + a^2*c)*e^6)*x^2 - (2560*c^3*d^
5*e - 8320*b*c^2*d^4*e^2 - 15015*a^2*b*e^6 + 9152*(b^2*c + a*c^2)*d^3*e^3 - 3432*(b^3 + 6*a*b*c)*d^2*e^4 + 120
12*(a*b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.21, size = 556, normalized size = 1.97 \[ \frac {2}{15015} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b e^{\left (-1\right )} + 3003 \, {\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^{2} e^{\left (-2\right )} + 3003 \, {\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^{2} c e^{\left (-2\right )} + 429 \, {\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^{3} e^{\left (-3\right )} + 2574 \, {\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 b c e^{\left (-3\right )} + 143 \, {\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^{2} c e^{\left (-4\right )} + 143 \, {\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 )} a c^{2} e^{\left (-4\right )} + 65 \, {\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 )} b c^{2} e^{\left (-5\right )} + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} e^{\left (-6\right )} + 15015 \, \sqrt {x e + d} a^{3}\right )} e^{\left (-1\right )} \]

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]

2/15015*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/
2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*e^(-2) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*
d^2)*a^2*c*e^(-2) + 429*(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*e^(-3) + 2574*(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*e^(-3) + 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*sqrt(x*e + d)*d^4)*b^2*c*e^(-4) + 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*sqrt(x*e + d)*d^4)*a*c^2*e^(-4) + 65*(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*e^(-5) + 5*(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*e^(-6
) + 15015*sqrt(x*e + d)*a^3)*e^(-1)

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maple [A]  time = 0.05, size = 495, normalized size = 1.76 \[ \frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}} \]

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]

2/15015*(e*x+d)^(1/2)*(1155*c^3*e^6*x^6+4095*b*c^2*e^6*x^5-1260*c^3*d*e^5*x^5+5005*a*c^2*e^6*x^4+5005*b^2*c*e^
6*x^4-4550*b*c^2*d*e^5*x^4+1400*c^3*d^2*e^4*x^4+12870*a*b*c*e^6*x^3-5720*a*c^2*d*e^5*x^3+2145*b^3*e^6*x^3-5720
*b^2*c*d*e^5*x^3+5200*b*c^2*d^2*e^4*x^3-1600*c^3*d^3*e^3*x^3+9009*a^2*c*e^6*x^2+9009*a*b^2*e^6*x^2-15444*a*b*c
*d*e^5*x^2+6864*a*c^2*d^2*e^4*x^2-2574*b^3*d*e^5*x^2+6864*b^2*c*d^2*e^4*x^2-6240*b*c^2*d^3*e^3*x^2+1920*c^3*d^
4*e^2*x^2+15015*a^2*b*e^6*x-12012*a^2*c*d*e^5*x-12012*a*b^2*d*e^5*x+20592*a*b*c*d^2*e^4*x-9152*a*c^2*d^3*e^3*x
+3432*b^3*d^2*e^4*x-9152*b^2*c*d^3*e^3*x+8320*b*c^2*d^4*e^2*x-2560*c^3*d^5*e*x+15015*a^3*e^6-30030*a^2*b*d*e^5
+24024*a^2*c*d^2*e^4+24024*a*b^2*d^2*e^4-41184*a*b*c*d^3*e^3+18304*a*c^2*d^4*e^2-6864*b^3*d^3*e^3+18304*b^2*c*
d^4*e^2-16640*b*c^2*d^5*e+5120*c^3*d^6)/e^7

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maxima [B]  time = 0.96, size = 525, normalized size = 1.86 \[ \frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + 3003 \, a^{2} {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )} + 143 \, a {\left (\frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )} + \frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]

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]

2/15015*(15015*sqrt(e*x + d)*a^3 + 3003*a^2*(5*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*b/e + (3*(e*x + d)^(5/2)
- 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c/e^2) + 143*a*(21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d +
15*sqrt(e*x + d)*d^2)*b^2/e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqr
t(e*x + d)*d^3)*b*c/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d
)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2/e^4) + 429*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(
3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3/e^3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/
2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^
(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^
5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(
7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e

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mupad [B]  time = 0.07, size = 297, normalized size = 1.05 \[ \frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{5\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{9\,e^7}+\frac {2\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(5/2)*(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))/(5*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d +
e*x)^(11/2))/(11*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d*e))/(9*e^7) + (2
*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)^3)/e^7 + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a
*c*e^2 - 10*b*c*d*e))/(7*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^2)/e^7

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sympy [A]  time = 158.27, size = 1406, normalized size = 4.99 \[ \text {result too large to display} \]

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)

[Out]

Piecewise(((-2*a**3*d/sqrt(d + e*x) - 2*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*a**2*b*d*(-d/sqrt(d + e*x)
- sqrt(d + e*x))/e - 6*a**2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*a**2*c*d*(d
**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*a**2*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt
(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 6*a*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x)
- (d + e*x)**(3/2)/3)/e**2 - 6*a*b**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d +
e*x)**(5/2)/5)/e**2 - 12*a*b*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)
**(5/2)/5)/e**3 - 12*a*b*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x
)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 6*a*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*
x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 6*a*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d
+ e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e
**4 - 2*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 2
*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x
)**(7/2)/7)/e**3 - 6*b**2*c*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d +
e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 6*b**2*c*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d +
e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 6*b*c**2*d*(-d*
*5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)
**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 6*b*c**2*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)*
*(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)
/e**5 - 2*c**3*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2
) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 2*c**3*(-d**7/sqrt(d +
e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*
d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((a**3*x + 3*a*
*2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a
**2*c + 3*a*b**2)/3)/sqrt(d), True))

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