3.11.26 \(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=79 \[ \frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]

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Rubi [A]  time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {693, 682} \begin {gather*} \frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac {2 \left (a+b x+c x^2\right )^{5/2}}{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)^(3/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7) + (4*(a + b*x + c*x^2)^(5/2))/(35*(b^2 - 4*a*c
)^2*d^8*(b + 2*c*x)^5)

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]

Rule 693

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx &=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {2 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx}{7 \left (b^2-4 a c\right ) d^2}\\ &=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac {4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 62, normalized size = 0.78 \begin {gather*} \frac {2 (a+x (b+c x))^{5/2} \left (4 c \left (2 c x^2-5 a\right )+7 b^2+8 b c x\right )}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*(a + x*(b + c*x))^(5/2)*(7*b^2 + 8*b*c*x + 4*c*(-5*a + 2*c*x^2)))/(35*(b^2 - 4*a*c)^2*d^8*(b + 2*c*x)^7)

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IntegrateAlgebraic [A]  time = 1.29, size = 154, normalized size = 1.95 \begin {gather*} \frac {2 \sqrt {a+b x+c x^2} \left (-20 a^3 c+7 a^2 b^2-32 a^2 b c x-32 a^2 c^2 x^2+14 a b^3 x+10 a b^2 c x^2-8 a b c^2 x^3-4 a c^3 x^4+7 b^4 x^2+22 b^3 c x^3+31 b^2 c^2 x^4+24 b c^3 x^5+8 c^4 x^6\right )}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x]

[Out]

(2*Sqrt[a + b*x + c*x^2]*(7*a^2*b^2 - 20*a^3*c + 14*a*b^3*x - 32*a^2*b*c*x + 7*b^4*x^2 + 10*a*b^2*c*x^2 - 32*a
^2*c^2*x^2 + 22*b^3*c*x^3 - 8*a*b*c^2*x^3 + 31*b^2*c^2*x^4 - 4*a*c^3*x^4 + 24*b*c^3*x^5 + 8*c^4*x^6))/(35*(b^2
- 4*a*c)^2*d^8*(b + 2*c*x)^7)

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fricas [B]  time = 9.68, size = 398, normalized size = 5.04 \begin {gather*} \frac {2 \, {\left (8 \, c^{4} x^{6} + 24 \, b c^{3} x^{5} + {\left (31 \, b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 7 \, a^{2} b^{2} - 20 \, a^{3} c + 2 \, {\left (11 \, b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (7 \, b^{4} + 10 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (7 \, a b^{3} - 16 \, a^{2} b c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{35 \, {\left (128 \, {\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \, {\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \, {\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \, {\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \, {\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \, {\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \, {\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x + {\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="fricas")

[Out]

2/35*(8*c^4*x^6 + 24*b*c^3*x^5 + (31*b^2*c^2 - 4*a*c^3)*x^4 + 7*a^2*b^2 - 20*a^3*c + 2*(11*b^3*c - 4*a*b*c^2)*
x^3 + (7*b^4 + 10*a*b^2*c - 32*a^2*c^2)*x^2 + 2*(7*a*b^3 - 16*a^2*b*c)*x)*sqrt(c*x^2 + b*x + a)/(128*(b^4*c^7
- 8*a*b^2*c^8 + 16*a^2*c^9)*d^8*x^7 + 448*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^8*x^6 + 672*(b^6*c^5 - 8*a*
b^4*c^6 + 16*a^2*b^2*c^7)*d^8*x^5 + 560*(b^7*c^4 - 8*a*b^5*c^5 + 16*a^2*b^3*c^6)*d^8*x^4 + 280*(b^8*c^3 - 8*a*
b^6*c^4 + 16*a^2*b^4*c^5)*d^8*x^3 + 84*(b^9*c^2 - 8*a*b^7*c^3 + 16*a^2*b^5*c^4)*d^8*x^2 + 14*(b^10*c - 8*a*b^8
*c^2 + 16*a^2*b^6*c^3)*d^8*x + (b^11 - 8*a*b^9*c + 16*a^2*b^7*c^2)*d^8)

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giac [B]  time = 0.90, size = 1003, normalized size = 12.70 \begin {gather*} \frac {560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{10} c^{\frac {11}{2}} + 2800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} b c^{5} + 6160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} b^{2} c^{\frac {9}{2}} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} a c^{\frac {11}{2}} + 7840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b^{3} c^{4} + 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} a b c^{5} + 6440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{4} c^{\frac {7}{2}} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a b^{2} c^{\frac {9}{2}} + 1120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} a^{2} c^{\frac {11}{2}} + 3640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{5} c^{3} + 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a b^{3} c^{4} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} a^{2} b c^{5} + 1484 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{6} c^{\frac {5}{2}} + 392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{4} c^{\frac {7}{2}} + 4032 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} b^{2} c^{\frac {9}{2}} + 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{3} c^{\frac {11}{2}} + 448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{7} c^{2} - 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{5} c^{3} + 2464 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b^{3} c^{4} + 448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{3} b c^{5} + 98 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{8} c^{\frac {3}{2}} - 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{6} c^{\frac {5}{2}} + 840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{4} c^{\frac {7}{2}} + 224 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} b^{2} c^{\frac {9}{2}} + 112 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{4} c^{\frac {11}{2}} + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{9} c - 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{7} c^{2} + 168 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{5} c^{3} + 112 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{4} b c^{5} + b^{10} \sqrt {c} - 6 \, a b^{8} c^{\frac {3}{2}} + 20 \, a^{2} b^{6} c^{\frac {5}{2}} - 24 \, a^{3} b^{4} c^{\frac {7}{2}} + 48 \, a^{4} b^{2} c^{\frac {9}{2}} - 16 \, a^{5} c^{\frac {11}{2}}}{280 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{7} c^{3} d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x, algorithm="giac")

[Out]

1/280*(560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*c^(11/2) + 2800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b*c^5
+ 6160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(9/2) + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(11/2
) + 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^4 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^5 +
6440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(7/2) + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(9
/2) + 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(11/2) + 3640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5
*c^3 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^4 + 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b
*c^5 + 1484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(5/2) + 392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^
4*c^(7/2) + 4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^(9/2) + 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*a^3*c^(11/2) + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^7*c^2 - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*a*b^5*c^3 + 2464*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c^4 + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*a^3*b*c^5 + 98*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c^(3/2) - 224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^2*a*b^6*c^(5/2) + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c^(7/2) + 224*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^3*b^2*c^(9/2) + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^(11/2) + 14*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*b^9*c - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^7*c^2 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^2*b^5*c^3 + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^5 + b^10*sqrt(c) - 6*a*b^8*c^(3/2) + 20*a^2
*b^6*c^(5/2) - 24*a^3*b^4*c^(7/2) + 48*a^4*b^2*c^(9/2) - 16*a^5*c^(11/2))/((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^3*d^8)

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maple [A]  time = 0.05, size = 70, normalized size = 0.89 \begin {gather*} -\frac {2 \left (-8 c^{2} x^{2}-8 b c x +20 a c -7 b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{35 \left (2 c x +b \right )^{7} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^8,x)

[Out]

-2/35*(-8*c^2*x^2-8*b*c*x+20*a*c-7*b^2)*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^7/d^8/(16*a^2*c^2-8*a*b^2*c+b^4)

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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)^(3/2)/(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 = 2.37, size = 1814, normalized size = 22.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^8,x)

[Out]

(((9*a*c - 2*b^2)/(70*c^2*d^8*(4*a*c - b^2)^3) - b^2/(280*c^2*d^8*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(
b + 2*c*x) - (((b*((b*((16*c^2*(7*a*c - b^2))/(35*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2)) - (6*b^2*c^2)/(
35*d^8*(4*a*c - b^2)^2*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (4*c*(6*b^3 - 28*a*b*c))/(35*d^8*(4*a*c - b^2)^2*(48
*a*c^3 - 12*b^2*c^2))))/(2*c) - (4*c*(6*a*b^2 - 26*a^2*c))/(35*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*((6*c^2*(12*a*c - b^2))/(7*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b
^2*c^2)) - (6*b^2*c^2)/(7*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (2*c*(7*b^3 - 36*a*b*c))/(7*d^8
*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (2*c*(7*a*b^2 - 32*a^2*c))/(7*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^2/(14*d^8*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) -
(8*a*c - b^2)/(14*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^2/(70*d^8*(4*a*c - b^2)^4) - (68*a*c^2 - 11*b^2*c)/(210*c*d^8*(4*a*c - b^2)^4)))/(2*c) - (15*b^3 - 68*a*b
*c)/(210*c*d^8*(4*a*c - b^2)^4)))/(2*c) + (15*a*b^2 - 64*a^2*c)/(210*c*d^8*(4*a*c - b^2)^4))*(a + b*x + c*x^2)
^(1/2))/(b + 2*c*x) + (((b*((b*((b*((16*c^3*(14*a*c - b^2))/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2)) -
(16*b^2*c^3)/(35*d^8*(4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (16*b*c^2*(21*a*c - 4*b^2))/(35*d^8*(
4*a*c - b^2)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (4*c*(52*a^2*c^2 - 6*b^4 + 16*a*b^2*c))/(35*d^8*(4*a*c - b^2
)^2*(64*a*c^3 - 16*b^2*c^2))))/(2*c) - (8*a*b*c*(13*a*c - 3*b^2))/(35*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*((4*c^3*(36*a*c + b^2))/(7*d^8*(4*a*c - b^2)*(96*a
*c^3 - 24*b^2*c^2)) - (16*b^2*c^3)/(7*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (2*c*(17*b^3*c - 10
8*a*b*c^2))/(7*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (2*c*(64*a^2*c^2 - 7*b^4 + 22*a*b^2*c))/(7
*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (2*c*(7*a*b^3 - 32*a^2*b*c))/(7*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*((b*((2*c*(136*a*c^3 - 14*b^2*c^2))/(105
*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2)) - (16*b^2*c^3)/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))
/(2*c) + (2*c*(41*b^3*c - 204*a*b*c^2))/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (2*c*(128*a
^2*c^2 - 15*b^4 + 38*a*b^2*c))/(105*d^8*(4*a*c - b^2)^3*(32*a*c^3 - 8*b^2*c^2))))/(2*c) + (2*c*(15*a*b^3 - 64*
a^2*b*c))/(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*
((4*c^2*(4*a*c + 2*b^2))/(d^8*(112*a*c^3 - 28*b^2*c^2)) - (6*b^2*c^2)/(d^8*(112*a*c^3 - 28*b^2*c^2))))/(2*c) -
(16*a*b*c^2)/(d^8*(112*a*c^3 - 28*b^2*c^2))))/(2*c) + (8*a^2*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 - (a + b*x + c*x^2)^(1/2)/(168*c^2*d^8*(4*a*c - b^2)^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 \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 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 {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}{d^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**8,x)

[Out]

(Integral(a*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*x*sq
rt(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 + 179
2*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*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))/d**8

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