3.11.41 \(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=39 \[ \frac {2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {682} \begin {gather*} \frac {2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 38, normalized size = 0.97 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + x*(b + c*x))^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)

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IntegrateAlgebraic [B]  time = 1.14, size = 117, normalized size = 3.00 \begin {gather*} \frac {2 \sqrt {a+b x+c x^2} \left (a^3+3 a^2 b x+3 a^2 c x^2+3 a b^2 x^2+6 a b c x^3+3 a c^2 x^4+b^3 x^3+3 b^2 c x^4+3 b c^2 x^5+c^3 x^6\right )}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]

[Out]

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

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fricas [B]  time = 9.71, size = 270, normalized size = 6.92 \begin {gather*} \frac {2 \, {\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{7 \, {\left (128 \, {\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{8} x^{7} + 448 \, {\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{8} x^{6} + 672 \, {\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{8} x^{5} + 560 \, {\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{8} x^{4} + 280 \, {\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{8} x^{3} + 84 \, {\left (b^{7} c^{2} - 4 \, a b^{5} c^{3}\right )} d^{8} x^{2} + 14 \, {\left (b^{8} c - 4 \, a b^{6} c^{2}\right )} d^{8} x + {\left (b^{9} - 4 \, a b^{7} c\right )} d^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="fricas")

[Out]

2/7*(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)
*x^2)*sqrt(c*x^2 + b*x + a)/(128*(b^2*c^7 - 4*a*c^8)*d^8*x^7 + 448*(b^3*c^6 - 4*a*b*c^7)*d^8*x^6 + 672*(b^4*c^
5 - 4*a*b^2*c^6)*d^8*x^5 + 560*(b^5*c^4 - 4*a*b^3*c^5)*d^8*x^4 + 280*(b^6*c^3 - 4*a*b^4*c^4)*d^8*x^3 + 84*(b^7
*c^2 - 4*a*b^5*c^3)*d^8*x^2 + 14*(b^8*c - 4*a*b^6*c^2)*d^8*x + (b^9 - 4*a*b^7*c)*d^8)

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giac [B]  time = 1.07, size = 1247, normalized size = 31.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="giac")

[Out]

1/448*(448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*c^(13/2) + 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b*c^6
+ 7392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^2*c^(11/2) + 12320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*
c^5 + 14000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(9/2) - 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b
^2*c^(11/2) + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(13/2) + 11648*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*b^5*c^4 - 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^5 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*a^2*b*c^6 + 7448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^(7/2) - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^6*a*b^4*c^(9/2) + 15680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*c^(11/2) + 3752*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*b^7*c^3 - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^5*c^4 + 15680*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a^2*b^3*c^5 + 1484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^8*c^(5/2) - 4984*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*a*b^6*c^(7/2) + 10304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(9/2) - 1344*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^(11/2) + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(13/2) + 4
48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*c^2 - 2128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^7*c^3 + 4928
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c^4 - 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*c^5 +
2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^6 + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^10*c^(3/2) -
616*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^8*c^(5/2) + 1736*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^6*
c^(7/2) - 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^4*c^(9/2) + 2016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a^4*b^2*c^(11/2) + 14*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^11*c - 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a*b^9*c^2 + 392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*c^3 - 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3
*b^5*c^4 + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3*c^5 + b^12*sqrt(c) - 10*a*b^10*c^(3/2) + 44*a^2*b^8
*c^(5/2) - 104*a^3*b^6*c^(7/2) + 144*a^4*b^4*c^(9/2) - 96*a^5*b^2*c^(11/2) + 64*a^6*c^(13/2))/((2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^7*c^4*d^8)

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maple [A]  time = 0.04, size = 38, normalized size = 0.97 \begin {gather*} -\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 \left (2 c x +b \right )^{7} \left (4 a c -b^{2}\right ) d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x)

[Out]

-2/7*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^7/d^8/(4*a*c-b^2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 3.70, size = 3088, normalized size = 79.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x)

[Out]

((a/(56*c^2*d^8*(4*a*c - b^2)^2) - b^2/(224*c^3*d^8*(4*a*c - b^2)^2))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (
((b*((b*((b*((b*((b*((4*c^4*(40*a*c + 11*b^2))/(7*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2)) - (24*b^2*c^4)/(7
*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c^3*(40*a*c - 3*b^2))/(7*d^8*(4*a*c - b^2)*(96*a*c
^3 - 24*b^2*c^2))))/(2*c) + (768*a^2*c^4 - 82*b^4*c^2 + 416*a*b^2*c^3)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^
2*c^2))))/(2*c) + (b*c*(7*b^4 - 1152*a^2*c^2 + 176*a*b^2*c))/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/
(2*c) + (7*b^6 + 480*a^3*c^3 + 216*a^2*b^2*c^2 - 98*a*b^4*c)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/
(2*c) - (7*a*b^5 - 84*a^2*b^3*c + 240*a^3*b*c^2)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*
x^2)^(1/2))/(b + 2*c*x)^6 + (((b*((b*((10*a*c - b^2)/(70*c*d^8*(4*a*c - b^2)^3) - (3*b^2)/(280*c*d^8*(4*a*c -
b^2)^3)))/(2*c) - (b*(5*a*c - b^2))/(35*c^2*d^8*(4*a*c - b^2)^3)))/(2*c) + (15*b^4 + 384*a^2*c^2 - 152*a*b^2*c
)/(1120*c^3*d^8*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((b*((10*c^3*(8*a*c + b^
2))/(7*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) - (10*b^2*c^3)/(7*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))
))/(2*c) - (20*b*c^2*(8*a*c - b^2))/(7*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (384*a^2*c^3 - 21*
b^4*c + 48*a*b^2*c^2)/(14*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (7*b^5 + 384*a^2*b*c^2 - 112*a*
b^3*c)/(14*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (7*a*b^4 + 240*a^3*c^2 - 84*a^2*b^2*c)/(14*d^8
*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^5 - (((b*((b*((b*((b*((24*c^3*(a
*c + b^2))/(d^8*(112*a*c^3 - 28*b^2*c^2)) - (10*b^2*c^3)/(d^8*(112*a*c^3 - 28*b^2*c^2))))/(2*c) - (4*c^2*(2*b^
3 + 12*a*b*c))/(d^8*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (24*a*c^2*(a*c + b^2))/(d^8*(112*a*c^3 - 28*b^2*c^2)))
)/(2*c) - (24*a^2*b*c^2)/(d^8*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (8*a^3*c^2)/(d^8*(112*a*c^3 - 28*b^2*c^2)))*
(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^7 - (((b*((b*((3*b^2*c)/(14*d^8*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) -
(3*c*(4*a*c + b^2))/(14*d^8*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (b*(12*a*c - b^2))/(14*d^8*(4*a*c
- b^2)*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (b^4 + 48*a^2*c^2 - 12*a*b^2*c)/(56*c*d^8*(4*a*c - b^2)*(48*a*c^3 -
12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 - (((b*((b*((b*((b*((24*a*c^4)/(7*d^8*(4*a*c - b^2)^2*(4
8*a*c^3 - 12*b^2*c^2)) - (2*b^2*c^3)/(7*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (8*b*c^2*(6*a*c
- b^2))/(7*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (b^4*c - 1184*a^2*c^3 + 232*a*b^2*c^2)/(70*
d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) - (47*b^5 + 1184*a^2*b*c^2 - 472*a*b^3*c)/(70*d^8*(4*a*c
- b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (47*a*b^4 + 960*a^3*c^2 - 424*a^2*b^2*c)/(70*d^8*(4*a*c - b^2)^2*(
48*a*c^3 - 12*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 + (((b*(b^2/(14*d^8*(4*a*c - b^2)*(32*a*c^3 -
8*b^2*c^2)) - (2*a*c + b^2)/(14*d^8*(4*a*c - b^2)*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (a*b)/(14*d^8*(4*a*c - b^2
)*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2 + (((b*((b*((b*((b*((b*((4*c^4*(24*a*c + b^2
))/(35*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (8*b^2*c^4)/(35*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2
))))/(2*c) - (2*b*c^3*(72*a*c - 11*b^2))/(21*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (26*b^4*c^2
- 1904*a^2*c^4 + 232*a*b^2*c^3)/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (b*c*(33*b^4 + 952
*a^2*c^2 - 356*a*b^2*c))/(35*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (43*b^6 + 1632*a^3*c^3 + 20
4*a^2*b^2*c^2 - 318*a*b^4*c)/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (43*a*b^5 - 374*a^2*b^
3*c + 816*a^3*b*c^2)/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2
+ (((b*((b*((b*((b*((b*((24*c^4*(10*a*c + b^2))/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)) - (24*b^2*c^4
)/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (8*b*c^3*(15*a*c - 2*b^2))/(7*d^8*(4*a*c - b^2)^2
*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2368*a^2*c^4 - 82*b^4*c^2 + 16*a*b^2*c^3)/(70*d^8*(4*a*c - b^2)^2*(64*a*c
^3 - 16*b^2*c^2))))/(2*c) - (3*b*c*(31*b^4 + 1184*a^2*c^2 - 392*a*b^2*c))/(70*d^8*(4*a*c - b^2)^2*(64*a*c^3 -
16*b^2*c^2))))/(2*c) + (47*b^6 + 1920*a^3*c^3 + 336*a^2*b^2*c^2 - 378*a*b^4*c)/(70*d^8*(4*a*c - b^2)^2*(64*a*c
^3 - 16*b^2*c^2))))/(2*c) - (47*a*b^5 - 424*a^2*b^3*c + 960*a^3*b*c^2)/(70*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*
b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^4 - (((b*((b*((b*((b*((c*(24*a*c - b^2))/(70*d^8*(4*a*c - b^2)
^4) - (b^2*c)/(42*d^8*(4*a*c - b^2)^4)))/(2*c) - (b*(72*a*c - 13*b^2))/(105*d^8*(4*a*c - b^2)^4)))/(2*c) + (13
*b^4*c + 952*a^2*c^3 - 260*a*b^2*c^2)/(420*c^2*d^8*(4*a*c - b^2)^4)))/(2*c) - (43*b^5 + 952*a^2*b*c^2 - 404*a*
b^3*c)/(420*c^2*d^8*(4*a*c - b^2)^4)))/(2*c) + (43*a*b^4 + 816*a^3*c^2 - 374*a^2*b^2*c)/(420*c^2*d^8*(4*a*c -
b^2)^4))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx}{d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**8,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**
4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(b**
2*x**2*sqrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x
**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(c**2*x**4*s
qrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 17
92*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(2*a*b*x*sqrt(a + b*
x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c*
*5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x
**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5
+ 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b
**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792
*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x))/d**8

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