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

Optimal. Leaf size=250 \[ \frac {8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac {4 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac {4 c^3 (d+e x)^{11/2}}{11 e^6} \]

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Rubi [A]  time = 0.13, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {771} \begin {gather*} \frac {8 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {2 (d+e x)^{5/2} (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 )}{5 e^6}+\frac {4 (d+e x)^{3/2} \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 )}{3 e^6}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {10 c^2 (d+e x)^{9/2} (2 c d-b e)}{9 e^6}+\frac {4 c^3 (d+e x)^{11/2}}{11 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^6 + (4*(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)^(3/2))/(3*e^6) - (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)^(5/2))/(5*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^6) - (10*
c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^6) + (4*c^3*(d + e*x)^(11/2))/(11*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 \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 ) \sqrt {d+e x}}{e^5}+\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)^{3/2}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{7/2}}{e^5}+\frac {2 c^3 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {4 \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)^{3/2}}{3 e^6}-\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)^{5/2}}{5 e^6}+\frac {8 c \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^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{9/2}}{9 e^6}+\frac {4 c^3 (d+e x)^{11/2}}{11 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 290, normalized size = 1.16 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-66 c e^2 \left (-35 a^2 e^2 (e x-2 d)-21 a b e \left (8 d^2-4 d e x+3 e^2 x^2\right )+6 b^2 \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+231 b e^3 \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+11 c^2 e \left (36 a e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+5 b \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )-10 c^3 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )}{3465 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-10*c^3*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5
) + 231*b*e^3*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 66*c*e^2*(-35*a^2*e^2
*(-2*d + e*x) - 21*a*b*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 6*b^2*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3))
+ 11*c^2*e*(36*a*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 5*b*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x
^2 - 40*d*e^3*x^3 + 35*e^4*x^4))))/(3465*e^6)

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IntegrateAlgebraic [A]  time = 0.17, size = 425, normalized size = 1.70 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3465 a^2 b e^5+2310 a^2 c e^4 (d+e x)-6930 a^2 c d e^4+2310 a b^2 e^4 (d+e x)-6930 a b^2 d e^4+20790 a b c d^2 e^3-13860 a b c d e^3 (d+e x)+4158 a b c e^3 (d+e x)^2-13860 a c^2 d^3 e^2+13860 a c^2 d^2 e^2 (d+e x)-8316 a c^2 d e^2 (d+e x)^2+1980 a c^2 e^2 (d+e x)^3+3465 b^3 d^2 e^3-2310 b^3 d e^3 (d+e x)+693 b^3 e^3 (d+e x)^2-13860 b^2 c d^3 e^2+13860 b^2 c d^2 e^2 (d+e x)-8316 b^2 c d e^2 (d+e x)^2+1980 b^2 c e^2 (d+e x)^3+17325 b c^2 d^4 e-23100 b c^2 d^3 e (d+e x)+20790 b c^2 d^2 e (d+e x)^2-9900 b c^2 d e (d+e x)^3+1925 b c^2 e (d+e x)^4-6930 c^3 d^5+11550 c^3 d^4 (d+e x)-13860 c^3 d^3 (d+e x)^2+9900 c^3 d^2 (d+e x)^3-3850 c^3 d (d+e x)^4+630 c^3 (d+e x)^5\right )}{3465 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-6930*c^3*d^5 + 17325*b*c^2*d^4*e - 13860*b^2*c*d^3*e^2 - 13860*a*c^2*d^3*e^2 + 3465*b^3*d^2
*e^3 + 20790*a*b*c*d^2*e^3 - 6930*a*b^2*d*e^4 - 6930*a^2*c*d*e^4 + 3465*a^2*b*e^5 + 11550*c^3*d^4*(d + e*x) -
23100*b*c^2*d^3*e*(d + e*x) + 13860*b^2*c*d^2*e^2*(d + e*x) + 13860*a*c^2*d^2*e^2*(d + e*x) - 2310*b^3*d*e^3*(
d + e*x) - 13860*a*b*c*d*e^3*(d + e*x) + 2310*a*b^2*e^4*(d + e*x) + 2310*a^2*c*e^4*(d + e*x) - 13860*c^3*d^3*(
d + e*x)^2 + 20790*b*c^2*d^2*e*(d + e*x)^2 - 8316*b^2*c*d*e^2*(d + e*x)^2 - 8316*a*c^2*d*e^2*(d + e*x)^2 + 693
*b^3*e^3*(d + e*x)^2 + 4158*a*b*c*e^3*(d + e*x)^2 + 9900*c^3*d^2*(d + e*x)^3 - 9900*b*c^2*d*e*(d + e*x)^3 + 19
80*b^2*c*e^2*(d + e*x)^3 + 1980*a*c^2*e^2*(d + e*x)^3 - 3850*c^3*d*(d + e*x)^4 + 1925*b*c^2*e*(d + e*x)^4 + 63
0*c^3*(d + e*x)^5))/(3465*e^6)

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fricas [A]  time = 0.41, size = 306, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (630 \, c^{3} e^{5} x^{5} - 2560 \, c^{3} d^{5} + 7040 \, b c^{2} d^{4} e + 3465 \, a^{2} b e^{5} - 6336 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 1848 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 4620 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 175 \, {\left (4 \, c^{3} d e^{4} - 11 \, b c^{2} e^{5}\right )} x^{4} + 20 \, {\left (40 \, c^{3} d^{2} e^{3} - 110 \, b c^{2} d e^{4} + 99 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \, {\left (320 \, c^{3} d^{3} e^{2} - 880 \, b c^{2} d^{2} e^{3} + 792 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - 231 \, {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \, {\left (640 \, c^{3} d^{4} e - 1760 \, b c^{2} d^{3} e^{2} + 1584 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 462 \, {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 1155 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(630*c^3*e^5*x^5 - 2560*c^3*d^5 + 7040*b*c^2*d^4*e + 3465*a^2*b*e^5 - 6336*(b^2*c + a*c^2)*d^3*e^2 + 18
48*(b^3 + 6*a*b*c)*d^2*e^3 - 4620*(a*b^2 + a^2*c)*d*e^4 - 175*(4*c^3*d*e^4 - 11*b*c^2*e^5)*x^4 + 20*(40*c^3*d^
2*e^3 - 110*b*c^2*d*e^4 + 99*(b^2*c + a*c^2)*e^5)*x^3 - 3*(320*c^3*d^3*e^2 - 880*b*c^2*d^2*e^3 + 792*(b^2*c +
a*c^2)*d*e^4 - 231*(b^3 + 6*a*b*c)*e^5)*x^2 + 2*(640*c^3*d^4*e - 1760*b*c^2*d^3*e^2 + 1584*(b^2*c + a*c^2)*d^2
*e^3 - 462*(b^3 + 6*a*b*c)*d*e^4 + 1155*(a*b^2 + a^2*c)*e^5)*x)*sqrt(e*x + d)/e^6

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giac [A]  time = 0.18, size = 421, normalized size = 1.68 \begin {gather*} \frac {2}{3465} \, {\left (2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b^{2} e^{\left (-1\right )} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c 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^{3} e^{\left (-2\right )} + 1386 \, {\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 c e^{\left (-2\right )} + 396 \, {\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} c e^{\left (-3\right )} + 396 \, {\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^{2} e^{\left (-3\right )} + 55 \, {\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^{2} e^{\left (-4\right )} + 10 \, {\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^{3} e^{\left (-5\right )} + 3465 \, \sqrt {x e + d} a^{2} b\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(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^2*e^(-1) + 2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a
^2*c*e^(-1) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^3*e^(-2) + 1386*(3*(x*e
+ d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*c*e^(-2) + 396*(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*c*e^(-3) + 396*(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^2*e^(-3) + 55*(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^2*e^(-4) + 10*(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^3*e^(-5) + 3465*sqrt(x*e + d)*a^2*b)*e^(-1)

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maple [A]  time = 0.05, size = 359, normalized size = 1.44 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (630 c^{3} e^{5} x^{5}+1925 b \,c^{2} e^{5} x^{4}-700 c^{3} d \,e^{4} x^{4}+1980 a \,c^{2} e^{5} x^{3}+1980 b^{2} c \,e^{5} x^{3}-2200 b \,c^{2} d \,e^{4} x^{3}+800 c^{3} d^{2} e^{3} x^{3}+4158 a b c \,e^{5} x^{2}-2376 a \,c^{2} d \,e^{4} x^{2}+693 b^{3} e^{5} x^{2}-2376 b^{2} c d \,e^{4} x^{2}+2640 b \,c^{2} d^{2} e^{3} x^{2}-960 c^{3} d^{3} e^{2} x^{2}+2310 a^{2} c \,e^{5} x +2310 a \,b^{2} e^{5} x -5544 a b c d \,e^{4} x +3168 a \,c^{2} d^{2} e^{3} x -924 b^{3} d \,e^{4} x +3168 b^{2} c \,d^{2} e^{3} x -3520 b \,c^{2} d^{3} e^{2} x +1280 c^{3} d^{4} e x +3465 a^{2} b \,e^{5}-4620 a^{2} c d \,e^{4}-4620 a \,b^{2} d \,e^{4}+11088 a b c \,d^{2} e^{3}-6336 a \,c^{2} d^{3} e^{2}+1848 b^{3} d^{2} e^{3}-6336 b^{2} c \,d^{3} e^{2}+7040 b \,c^{2} d^{4} e -2560 c^{3} d^{5}\right )}{3465 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(1/2)*(630*c^3*e^5*x^5+1925*b*c^2*e^5*x^4-700*c^3*d*e^4*x^4+1980*a*c^2*e^5*x^3+1980*b^2*c*e^5*x
^3-2200*b*c^2*d*e^4*x^3+800*c^3*d^2*e^3*x^3+4158*a*b*c*e^5*x^2-2376*a*c^2*d*e^4*x^2+693*b^3*e^5*x^2-2376*b^2*c
*d*e^4*x^2+2640*b*c^2*d^2*e^3*x^2-960*c^3*d^3*e^2*x^2+2310*a^2*c*e^5*x+2310*a*b^2*e^5*x-5544*a*b*c*d*e^4*x+316
8*a*c^2*d^2*e^3*x-924*b^3*d*e^4*x+3168*b^2*c*d^2*e^3*x-3520*b*c^2*d^3*e^2*x+1280*c^3*d^4*e*x+3465*a^2*b*e^5-46
20*a^2*c*d*e^4-4620*a*b^2*d*e^4+11088*a*b*c*d^2*e^3-6336*a*c^2*d^3*e^2+1848*b^3*d^2*e^3-6336*b^2*c*d^3*e^2+704
0*b*c^2*d^4*e-2560*c^3*d^5)/e^6

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maxima [A]  time = 0.53, size = 308, normalized size = 1.23 \begin {gather*} \frac {2 \, {\left (630 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 1925 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 1980 \, {\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 {7}{2}} - 693 \, {\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 {5}{2}} + 2310 \, {\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 {3}{2}} - 3465 \, {\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 )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(630*(e*x + d)^(11/2)*c^3 - 1925*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2) + 1980*(5*c^3*d^2 - 5*b*c^2*d*e +
(b^2*c + a*c^2)*e^2)*(e*x + d)^(7/2) - 693*(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)^(5/2) + 2310*(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)^(3/2) - 3465*(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)*sqrt(e*x + d))/e^6

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mupad [B]  time = 1.85, size = 267, normalized size = 1.07 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{3\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{7\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{5\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(3/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 -
40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(3*e^6) + (4*c^3*(d + e*x)^(11/2))/(11*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d +
e*x)^(9/2))/(9*e^6) + ((d + e*x)^(7/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(7*e^6) + (2*
(b*e - 2*c*d)*(d + e*x)^(5/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(5*e^6) + (2*(b*e - 2*c*d)*(d +
e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)^2)/e^6

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sympy [A]  time = 121.36, size = 1025, normalized size = 4.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a**2*b*d/sqrt(d + e*x) - 2*a**2*b*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 4*a**2*c*d*(-d/sqrt(d +
e*x) - sqrt(d + e*x))/e - 4*a**2*c*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 4*a*b**2*
d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e - 4*a*b**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3
)/e - 12*a*b*c*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 12*a*b*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 - 8*a*c**2*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 - 8*a*c**2*(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 - 2*b**3*d*(d**2
/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*b**3*(-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 - 8*b**2*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 - 8*b**2*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 - 10*b*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 - 10*b*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 - 4*c**3*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 - 4*c**3*(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)/e, Ne(e, 0)), ((a + b*x + c*x**2)**3/(3*sqrt(d)), Tr
ue))

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