3.24.54 \(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^8} \, dx\) [2354]

Optimal. Leaf size=510 \[ -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2048 \left (c d^2-b d e+a e^2\right )^{11/2}} \]

[Out]

1/128*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(a*e
^2-b*d*e+c*d^2)^4/(e*x+d)^4-1/7*e*(c*x^2+b*x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^7-3/28*e*(-b*e+2*c*d)*(c*x^2
+b*x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^6-1/280*e*(68*c^2*d^2+21*b^2*e^2-4*c*e*(4*a*e+17*b*d))*(c*x^2+b*x+
a)^(5/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^5+3/2048*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*
b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2
)^(11/2)-3/1024*(-4*a*c+b^2)*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(
c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^5/(e*x+d)^2

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Rubi [A]
time = 0.52, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {758, 848, 820, 734, 738, 212} \begin {gather*} -\frac {e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac {3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*
Sqrt[a + b*x + c*x^2])/(1024*(c*d^2 - b*d*e + a*e^2)^5*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 -
4*c*e*(2*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(128*(c*d^2 - b*d*e + a*e^2)^4*(
d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(7*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^7) - (3*e*(2*c*d - b*e)*(a + b*
x + c*x^2)^(5/2))/(28*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^6) - (e*(68*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(17*b*d + 4
*a*e))*(a + b*x + c*x^2)^(5/2))/(280*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5) + (3*(b^2 - 4*a*c)^2*(2*c*d - b*e)
*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e +
a*e^2]*Sqrt[a + b*x + c*x^2])])/(2048*(c*d^2 - b*d*e + a*e^2)^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

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

Rule 820

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

Rule 848

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {\int \frac {\left (\frac {1}{2} (-14 c d+9 b e)+2 c e x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx}{7 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}+\frac {\int \frac {\left (\frac {3}{4} \left (56 c^2 d^2+21 b^2 e^2-2 c e (31 b d+8 a e)\right )-\frac {9}{2} c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{42 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{256 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2048 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2048 \left (c d^2-b d e+a e^2\right )^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 15.22, size = 430, normalized size = 0.84 \begin {gather*} -\frac {\frac {e (a+x (b+c x))^{5/2}}{(d+e x)^7}+\frac {3 e (2 c d-b e) (a+x (b+c x))^{5/2}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^6}+\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) (a+x (b+c x))^{5/2}}{40 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^5}+\frac {7 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{256 \left (c d^2+e (-b d+a e)\right )^3}}{7 \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*((e*(a + x*(b + c*x))^(5/2))/(d + e*x)^7 + (3*e*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(4*(c*d^2 + e*(-(b
*d) + a*e))*(d + e*x)^6) + (e*(68*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(17*b*d + 4*a*e))*(a + x*(b + c*x))^(5/2))/(40*
(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^5) + (7*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*((2
*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b + c
*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(
b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*
d) + a*e))^(3/2)))))/(256*(c*d^2 + e*(-(b*d) + a*e))^3))/(c*d^2 + e*(-(b*d) + a*e))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15719\) vs. \(2(480)=960\).
time = 0.82, size = 15720, normalized size = 30.82

method	result	size
default	\(\text {Expression too large to display}\)	\(15720\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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)/(e*x+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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22291 vs. \(2 (502) = 1004\).
time = 8.92, size = 22291, normalized size = 43.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1024*(16*b^4*c^3*d^3 - 128*a*b^2*c^4*d^3 + 256*a^2*c^5*d^3 - 24*b^5*c^2*d^2*e + 192*a*b^3*c^3*d^2*e - 384*a^
2*b*c^4*d^2*e + 14*b^6*c*d*e^2 - 120*a*b^4*c^2*d*e^2 + 288*a^2*b^2*c^3*d*e^2 - 128*a^3*c^4*d*e^2 - 3*b^7*e^3 +
28*a*b^5*c*e^3 - 80*a^2*b^3*c^2*e^3 + 64*a^3*b*c^3*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt
(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^5*d^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 + 5*a*c^4*d^8*e^2 - 10*b^
3*c^2*d^7*e^3 - 20*a*b*c^3*d^7*e^3 + 5*b^4*c*d^6*e^4 + 30*a*b^2*c^2*d^6*e^4 + 10*a^2*c^3*d^6*e^4 - b^5*d^5*e^5
- 20*a*b^3*c*d^5*e^5 - 30*a^2*b*c^2*d^5*e^5 + 5*a*b^4*d^4*e^6 + 30*a^2*b^2*c*d^4*e^6 + 10*a^3*c^2*d^4*e^6 - 1
0*a^2*b^3*d^3*e^7 - 20*a^3*b*c*d^3*e^7 + 10*a^3*b^2*d^2*e^8 + 5*a^4*c*d^2*e^8 - 5*a^4*b*d*e^9 + a^5*e^10)*sqrt
(-c*d^2 + b*d*e - a*e^2)) + 1/35840*(114688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(19/2)*d^12*e + 32768*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^7*c^10*d^13 + 172032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*c^9*d^11*e^2 + 2457
60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^9*d^12*e + 114688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c^(19/2
)*d^13 + 143360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*c^(17/2)*d^10*e^3 + 86016*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^8*b*c^(17/2)*d^11*e^2 + 86016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(17/2)*d^12*e - 114688*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(19/2)*d^12*e + 172032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^9*d^13
- 229376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b*c^8*d^10*e^3 - 561152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b
^2*c^8*d^11*e^2 - 16384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^9*d^11*e^2 - 229376*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*b^3*c^8*d^12*e - 344064*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^9*d^12*e + 143360*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^4*b^3*c^(17/2)*d^13 - 716800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b*c^(15/2)*d^9*e
^4 - 931840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^(15/2)*d^10*e^3 + 716800*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^8*a*c^(17/2)*d^10*e^3 - 931840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(15/2)*d^11*e^2 + 630784*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*c^(17/2)*d^11*e^2 - 322560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^
(15/2)*d^12*e - 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(17/2)*d^12*e + 71680*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*b^4*c^8*d^13 - 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2*c^7*d^9*e^4 + 860160*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^9*a*c^8*d^9*e^4 - 706560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^7*d^10*e^3
+ 2957312*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^8*d^10*e^3 - 591360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^5*b^4*c^7*d^11*e^2 + 1892352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^8*d^11*e^2 + 172032*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*a^2*c^9*d^11*e^2 - 193536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^7*d^12*e - 28672
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^8*d^12*e + 21504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(
15/2)*d^13 + 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^2*c^(13/2)*d^8*e^5 + 716800*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^10*a*c^(15/2)*d^8*e^5 - 645120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(13/2)*d^9*e^4 + 28
6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(15/2)*d^9*e^4 + 188160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6
*b^4*c^(13/2)*d^10*e^3 + 3799040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(15/2)*d^10*e^3 - 716800*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(17/2)*d^10*e^3 - 137984*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^5*c^(13
/2)*d^11*e^2 + 2293760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(15/2)*d^11*e^2 + 430080*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a^2*b*c^(17/2)*d^11*e^2 - 62720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(13/2)*d^12*
e - 107520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(15/2)*d^12*e + 3584*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*b^6*c^7*d^13 + 4587520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^6*d^8*e^5 - 286720*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^9*a*b*c^7*d^8*e^5 + 775936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^6*d^9*e^4 - 4302848*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^7*d^9*e^4 - 5033984*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^8*d
^9*e^4 + 627200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^6*d^10*e^3 + 1361920*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*a*b^3*c^7*d^10*e^3 - 3096576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^8*d^10*e^3 + 25088*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^6*d^11*e^2 + 1469440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*c^7*d^11
*e^2 + 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^8*d^11*e^2 - 10752*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*b^7*c^6*d^12*e - 21504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^7*d^12*e + 256*b^7*c^(13/2)*d^13 - 1
433600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^3*c^(11/2)*d^7*e^6 - 2867200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^10*a*b*c^(13/2)*d^7*e^6 + 5565056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^(11/2)*d^8*e^5 + 2573312*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(13/2...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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