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

Optimal. Leaf size=248 \[ \frac {8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac {4 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt {d+e x}}-\frac {10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac {4 c^3 (d+e x)^{9/2}}{9 e^6} \]

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Rubi [A]  time = 0.13, antiderivative size = 248, 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)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (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 )}{3 e^6}+\frac {4 \sqrt {d+e x} \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 )}{e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt {d+e x}}-\frac {10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac {4 c^3 (d+e x)^{9/2}}{9 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.34, size = 287, normalized size = 1.16 \begin {gather*} \frac {252 c e^2 \left (5 a^2 e^2 (2 d+e x)+5 a b e \left (-8 d^2-4 d e x+e^2 x^2\right )+2 b^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-210 b e^3 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )-18 c^2 e \left (5 b \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-28 a e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+20 c^3 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )}{315 e^6 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*c^3*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) - 210*b*e^3*(3*a^
2*e^2 - 6*a*b*e*(2*d + e*x) + b^2*(8*d^2 + 4*d*e*x - e^2*x^2)) + 252*c*e^2*(5*a^2*e^2*(2*d + e*x) + 5*a*b*e*(-
8*d^2 - 4*d*e*x + e^2*x^2) + 2*b^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 18*c^2*e*(-28*a*e*(16*d^3 +
8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 5*b*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)))/
(315*e^6*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.17, size = 425, normalized size = 1.71 \begin {gather*} \frac {2 \left (-315 a^2 b e^5+630 a^2 c e^4 (d+e x)+630 a^2 c d e^4+630 a b^2 e^4 (d+e x)+630 a b^2 d e^4-1890 a b c d^2 e^3-3780 a b c d e^3 (d+e x)+630 a b c e^3 (d+e x)^2+1260 a c^2 d^3 e^2+3780 a c^2 d^2 e^2 (d+e x)-1260 a c^2 d e^2 (d+e x)^2+252 a c^2 e^2 (d+e x)^3-315 b^3 d^2 e^3-630 b^3 d e^3 (d+e x)+105 b^3 e^3 (d+e x)^2+1260 b^2 c d^3 e^2+3780 b^2 c d^2 e^2 (d+e x)-1260 b^2 c d e^2 (d+e x)^2+252 b^2 c e^2 (d+e x)^3-1575 b c^2 d^4 e-6300 b c^2 d^3 e (d+e x)+3150 b c^2 d^2 e (d+e x)^2-1260 b c^2 d e (d+e x)^3+225 b c^2 e (d+e x)^4+630 c^3 d^5+3150 c^3 d^4 (d+e x)-2100 c^3 d^3 (d+e x)^2+1260 c^3 d^2 (d+e x)^3-450 c^3 d (d+e x)^4+70 c^3 (d+e x)^5\right )}{315 e^6 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(630*c^3*d^5 - 1575*b*c^2*d^4*e + 1260*b^2*c*d^3*e^2 + 1260*a*c^2*d^3*e^2 - 315*b^3*d^2*e^3 - 1890*a*b*c*d^
2*e^3 + 630*a*b^2*d*e^4 + 630*a^2*c*d*e^4 - 315*a^2*b*e^5 + 3150*c^3*d^4*(d + e*x) - 6300*b*c^2*d^3*e*(d + e*x
) + 3780*b^2*c*d^2*e^2*(d + e*x) + 3780*a*c^2*d^2*e^2*(d + e*x) - 630*b^3*d*e^3*(d + e*x) - 3780*a*b*c*d*e^3*(
d + e*x) + 630*a*b^2*e^4*(d + e*x) + 630*a^2*c*e^4*(d + e*x) - 2100*c^3*d^3*(d + e*x)^2 + 3150*b*c^2*d^2*e*(d
+ e*x)^2 - 1260*b^2*c*d*e^2*(d + e*x)^2 - 1260*a*c^2*d*e^2*(d + e*x)^2 + 105*b^3*e^3*(d + e*x)^2 + 630*a*b*c*e
^3*(d + e*x)^2 + 1260*c^3*d^2*(d + e*x)^3 - 1260*b*c^2*d*e*(d + e*x)^3 + 252*b^2*c*e^2*(d + e*x)^3 + 252*a*c^2
*e^2*(d + e*x)^3 - 450*c^3*d*(d + e*x)^4 + 225*b*c^2*e*(d + e*x)^4 + 70*c^3*(d + e*x)^5))/(315*e^6*Sqrt[d + e*
x])

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fricas [A]  time = 0.40, size = 316, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (70 \, c^{3} e^{5} x^{5} + 2560 \, c^{3} d^{5} - 5760 \, b c^{2} d^{4} e - 315 \, a^{2} b e^{5} + 4032 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 840 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 1260 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 25 \, {\left (4 \, c^{3} d e^{4} - 9 \, b c^{2} e^{5}\right )} x^{4} + 4 \, {\left (40 \, c^{3} d^{2} e^{3} - 90 \, b c^{2} d e^{4} + 63 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - {\left (320 \, c^{3} d^{3} e^{2} - 720 \, b c^{2} d^{2} e^{3} + 504 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - 105 \, {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \, {\left (640 \, c^{3} d^{4} e - 1440 \, b c^{2} d^{3} e^{2} + 1008 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 210 \, {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 315 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\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)^(3/2),x, algorithm="fricas")

[Out]

2/315*(70*c^3*e^5*x^5 + 2560*c^3*d^5 - 5760*b*c^2*d^4*e - 315*a^2*b*e^5 + 4032*(b^2*c + a*c^2)*d^3*e^2 - 840*(
b^3 + 6*a*b*c)*d^2*e^3 + 1260*(a*b^2 + a^2*c)*d*e^4 - 25*(4*c^3*d*e^4 - 9*b*c^2*e^5)*x^4 + 4*(40*c^3*d^2*e^3 -
90*b*c^2*d*e^4 + 63*(b^2*c + a*c^2)*e^5)*x^3 - (320*c^3*d^3*e^2 - 720*b*c^2*d^2*e^3 + 504*(b^2*c + a*c^2)*d*e
^4 - 105*(b^3 + 6*a*b*c)*e^5)*x^2 + 2*(640*c^3*d^4*e - 1440*b*c^2*d^3*e^2 + 1008*(b^2*c + a*c^2)*d^2*e^3 - 210
*(b^3 + 6*a*b*c)*d*e^4 + 315*(a*b^2 + a^2*c)*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

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giac [B]  time = 0.21, size = 460, normalized size = 1.85 \begin {gather*} \frac {2}{315} \, {\left (70 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} e^{48} - 450 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e^{48} + 1260 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{48} - 2100 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{48} + 3150 \, \sqrt {x e + d} c^{3} d^{4} e^{48} + 225 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{49} - 1260 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{49} + 3150 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{49} - 6300 \, \sqrt {x e + d} b c^{2} d^{3} e^{49} + 252 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{50} + 252 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} e^{50} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{50} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{50} + 3780 \, \sqrt {x e + d} b^{2} c d^{2} e^{50} + 3780 \, \sqrt {x e + d} a c^{2} d^{2} e^{50} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{51} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c e^{51} - 630 \, \sqrt {x e + d} b^{3} d e^{51} - 3780 \, \sqrt {x e + d} a b c d e^{51} + 630 \, \sqrt {x e + d} a b^{2} e^{52} + 630 \, \sqrt {x e + d} a^{2} c e^{52}\right )} e^{\left (-54\right )} + \frac {2 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/315*(70*(x*e + d)^(9/2)*c^3*e^48 - 450*(x*e + d)^(7/2)*c^3*d*e^48 + 1260*(x*e + d)^(5/2)*c^3*d^2*e^48 - 2100
*(x*e + d)^(3/2)*c^3*d^3*e^48 + 3150*sqrt(x*e + d)*c^3*d^4*e^48 + 225*(x*e + d)^(7/2)*b*c^2*e^49 - 1260*(x*e +
d)^(5/2)*b*c^2*d*e^49 + 3150*(x*e + d)^(3/2)*b*c^2*d^2*e^49 - 6300*sqrt(x*e + d)*b*c^2*d^3*e^49 + 252*(x*e +
d)^(5/2)*b^2*c*e^50 + 252*(x*e + d)^(5/2)*a*c^2*e^50 - 1260*(x*e + d)^(3/2)*b^2*c*d*e^50 - 1260*(x*e + d)^(3/2
)*a*c^2*d*e^50 + 3780*sqrt(x*e + d)*b^2*c*d^2*e^50 + 3780*sqrt(x*e + d)*a*c^2*d^2*e^50 + 105*(x*e + d)^(3/2)*b
^3*e^51 + 630*(x*e + d)^(3/2)*a*b*c*e^51 - 630*sqrt(x*e + d)*b^3*d*e^51 - 3780*sqrt(x*e + d)*a*b*c*d*e^51 + 63
0*sqrt(x*e + d)*a*b^2*e^52 + 630*sqrt(x*e + d)*a^2*c*e^52)*e^(-54) + 2*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^
3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5)*e^(-6)/sq
rt(x*e + d)

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maple [A]  time = 0.05, size = 359, normalized size = 1.45 \begin {gather*} -\frac {2 \left (-70 c^{3} e^{5} x^{5}-225 b \,c^{2} e^{5} x^{4}+100 c^{3} d \,e^{4} x^{4}-252 a \,c^{2} e^{5} x^{3}-252 b^{2} c \,e^{5} x^{3}+360 b \,c^{2} d \,e^{4} x^{3}-160 c^{3} d^{2} e^{3} x^{3}-630 a b c \,e^{5} x^{2}+504 a \,c^{2} d \,e^{4} x^{2}-105 b^{3} e^{5} x^{2}+504 b^{2} c d \,e^{4} x^{2}-720 b \,c^{2} d^{2} e^{3} x^{2}+320 c^{3} d^{3} e^{2} x^{2}-630 a^{2} c \,e^{5} x -630 a \,b^{2} e^{5} x +2520 a b c d \,e^{4} x -2016 a \,c^{2} d^{2} e^{3} x +420 b^{3} d \,e^{4} x -2016 b^{2} c \,d^{2} e^{3} x +2880 b \,c^{2} d^{3} e^{2} x -1280 c^{3} d^{4} e x +315 a^{2} b \,e^{5}-1260 a^{2} c d \,e^{4}-1260 a \,b^{2} d \,e^{4}+5040 a b c \,d^{2} e^{3}-4032 a \,c^{2} d^{3} e^{2}+840 b^{3} d^{2} e^{3}-4032 b^{2} c \,d^{3} e^{2}+5760 b \,c^{2} d^{4} e -2560 c^{3} d^{5}\right )}{315 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/315/(e*x+d)^(1/2)*(-70*c^3*e^5*x^5-225*b*c^2*e^5*x^4+100*c^3*d*e^4*x^4-252*a*c^2*e^5*x^3-252*b^2*c*e^5*x^3+
360*b*c^2*d*e^4*x^3-160*c^3*d^2*e^3*x^3-630*a*b*c*e^5*x^2+504*a*c^2*d*e^4*x^2-105*b^3*e^5*x^2+504*b^2*c*d*e^4*
x^2-720*b*c^2*d^2*e^3*x^2+320*c^3*d^3*e^2*x^2-630*a^2*c*e^5*x-630*a*b^2*e^5*x+2520*a*b*c*d*e^4*x-2016*a*c^2*d^
2*e^3*x+420*b^3*d*e^4*x-2016*b^2*c*d^2*e^3*x+2880*b*c^2*d^3*e^2*x-1280*c^3*d^4*e*x+315*a^2*b*e^5-1260*a^2*c*d*
e^4-1260*a*b^2*d*e^4+5040*a*b*c*d^2*e^3-4032*a*c^2*d^3*e^2+840*b^3*d^2*e^3-4032*b^2*c*d^3*e^2+5760*b*c^2*d^4*e
-2560*c^3*d^5)/e^6

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maxima [A]  time = 0.64, size = 316, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (\frac {70 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} - 225 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 252 \, {\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 {5}{2}} - 105 \, {\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 {3}{2}} + 630 \, {\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 )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\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} e^{5}}\right )}}{315 \, e} \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)^(3/2),x, algorithm="maxima")

[Out]

2/315*((70*(e*x + d)^(9/2)*c^3 - 225*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 252*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2
*c + a*c^2)*e^2)*(e*x + d)^(5/2) - 105*(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)^(3/2) + 630*(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)*sqrt(e*x + d))/e^5 + 315*(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^5))/e

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mupad [B]  time = 0.07, size = 333, normalized size = 1.34 \begin {gather*} \frac {\sqrt {d+e\,x}\,\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 )}{e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\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 )}{5\,e^6}+\frac {-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5}{e^6\,\sqrt {d+e\,x}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{3\,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)^(3/2),x)

[Out]

((d + e*x)^(1/2)*(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))/e^6 + (4*c^3*(d + e*x)^(9/2))/(9*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e*x)^
(7/2))/(7*e^6) + ((d + e*x)^(5/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(5*e^6) + (4*c^3*d^
5 - 2*a^2*b*e^5 - 2*b^3*d^2*e^3 + 8*a*c^2*d^3*e^2 + 8*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 - 10*b*c^2
*d^4*e - 12*a*b*c*d^2*e^3)/(e^6*(d + e*x)^(1/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(b^2*e^2 + 10*c^2*d^2 + 6*
a*c*e^2 - 10*b*c*d*e))/(3*e^6)

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sympy [A]  time = 90.27, size = 316, normalized size = 1.27 \begin {gather*} \frac {4 c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (10 b c^{2} e - 20 c^{3} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (8 a c^{2} e^{2} + 8 b^{2} c e^{2} - 40 b c^{2} d e + 40 c^{3} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \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 )}{e^{6}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

4*c**3*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(10*b*c**2*e - 20*c**3*d)/(7*e**6) + (d + e*x)**(5/2)*(8*a
*c**2*e**2 + 8*b**2*c*e**2 - 40*b*c**2*d*e + 40*c**3*d**2)/(5*e**6) + (d + e*x)**(3/2)*(12*a*b*c*e**3 - 24*a*c
**2*d*e**2 + 2*b**3*e**3 - 24*b**2*c*d*e**2 + 60*b*c**2*d**2*e - 40*c**3*d**3)/(3*e**6) + sqrt(d + e*x)*(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)/e**6 - 2*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(e**6*sqrt(d + e*x))

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