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

Optimal. Leaf size=236 \[ \frac {3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 e^4 \sqrt {a e^2-b d e+c d^2}}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^4}+\frac {3 \sqrt {a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

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Rubi [A]  time = 0.25, antiderivative size = 236, 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 = {732, 812, 843, 621, 206, 724} \begin {gather*} \frac {3 \left (-4 c e (2 b d-a e)+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 )}{8 e^4 \sqrt {a e^2-b d e+c d^2}}+\frac {3 \sqrt {a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(4*e^3*(d + e*x)) - (a + b*x + c*x^2)^(3/2)/(2*e*(d + e*x)^2
) - (3*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*e^4) + (3*(8*c^2*d^2 +
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])])/(8*e^4*Sqrt[c*d^2 - b*d*e + a*e^2])

Rule 206

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

Rule 621

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

Rule 724

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 732

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

Rule 812

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^2} \, dx}{4 e}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {a+b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \int \frac {4 b c d-b^2 e-4 a c e+4 c (2 c d-b e) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 e^3}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {a+b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {(3 c (2 c d-b e)) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 e^4}+\frac {\left (3 \left (8 c^2 d^2+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}{8 e^4}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {a+b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {(3 c (2 c d-b e)) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{e^4}-\frac {\left (3 \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )\right ) \operatorname {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 )}{4 e^4}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {a+b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^4}+\frac {3 \left (8 c^2 d^2+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 )}{8 e^4 \sqrt {c d^2-b d e+a e^2}}\\ \end {align*}

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Mathematica [A]  time = 0.89, size = 311, normalized size = 1.32 \begin {gather*} \frac {\frac {3 \left (-\frac {2 e \sqrt {a+x (b+c x)} \left (-c e (2 a e-5 b d+b e x)-b^2 e^2+2 c^2 d (e x-2 d)\right )}{e (a e-b d)+c d^2}-\frac {\left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\sqrt {e (a e-b d)+c d^2}}-4 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2 e^3}+\frac {3 (a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x) \left (e (a e-b d)+c d^2\right )}-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^2}}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 83.64, size = 7971, normalized size = 33.78 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [B]  time = 20.43, size = 2711, normalized size = 11.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(12*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a
*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)*sqr
t(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 3*(8*c^2*d^4 - 8*
b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8
*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*
d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b
*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2))
- 4*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d
*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b
*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a
*d*e^7)*x), 1/16*(24*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d
^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e
^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 3*(8*c^2*d^4
- 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e
- 8*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a
*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a
)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^
2)) + 4*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b
*c*d*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2
+ b*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6
+ a*d*e^7)*x), 1/8*(3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2
+ 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arct
an(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e
+ a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 6*(2*c^2*d^5 - 3*b*c*d^4*e
- a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 +
2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2
- 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 2*(12*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2
*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4 + a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3
- 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e
^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a*d*e^7)*x), 1/8*(3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4
*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (b^2
+ 4*a*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)
*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 -
b^2*d*e + a*b*e^2)*x)) + 12*(2*c^2*d^5 - 3*b*c*d^4*e - a*b*d^2*e^3 + (b^2 + 2*a*c)*d^3*e^2 + (2*c^2*d^3*e^2 -
3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 - a*b*d*e^4 + (b^2 + 2*a*
c)*d^2*e^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(12
*c^2*d^4*e - 15*b*c*d^3*e^2 - a*b*d*e^4 - 2*a^2*e^5 + (3*b^2 + 10*a*c)*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4 +
a*c*e^5)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 - 5*a*b*e^5 + (5*b^2 + 18*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a)
)/(c*d^4*e^4 - b*d^3*e^5 + a*d^2*e^6 + (c*d^2*e^6 - b*d*e^7 + a*e^8)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6 + a*d*e^7)
*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.74Unable to divide, perhaps due to rounding error%%%{1,[6,0,7,0,0]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-
1,[1]%%%}]%%},[5,0,6,1,0]%%%}+%%%{3,[4,1,6,1,0]%%%}+%%%{-3,[4,0,7,0,1]%%%}+%%%{%%%{12,[1]%%%},[4,0,5,2,0]%%%}+
%%%{%%{[-12,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,5,2,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,6,1,1]%%%}+%
%%{%%{[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,4,3,0]%%%}+%%%{3,[2,2,5,2,0]%%%}+%%%{-6,[2,1,6,1,1]%%%}+
%%%{%%%{12,[1]%%%},[2,1,4,3,0]%%%}+%%%{3,[2,0,7,0,2]%%%}+%%%{%%%{-12,[1]%%%},[2,0,5,2,1]%%%}+%%%{%%{[-6,0]:[1,
0,%%%{-1,[1]%%%}]%%},[1,2,4,3,0]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,5,2,1]%%%}+%%%{%%{[-6,0]:[1,0,
%%%{-1,[1]%%%}]%%},[1,0,6,1,2]%%%}+%%%{1,[0,3,4,3,0]%%%}+%%%{-3,[0,2,5,2,1]%%%}+%%%{3,[0,1,6,1,2]%%%}+%%%{-1,[
0,0,7,0,3]%%%} / %%%{%%{poly1[%%%{1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,3,0,0]%%%}+%%%{%%%{-6,[2]%%%},[5,0
,2,1,0]%%%}+%%%{%%{[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,1,2,1,0]%%%}+%%%{%%{poly1[%%%{-3,[1]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[4,0,3,0,1]%%%}+%%%{%%{poly1[%%%{12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1,2,0]%%%}+%
%%{%%%{-12,[2]%%%},[3,1,1,2,0]%%%}+%%%{%%%{12,[2]%%%},[3,0,2,1,1]%%%}+%%%{%%%{-8,[3]%%%},[3,0,0,3,0]%%%}+%%%{%
%{[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,2,1,2,0]%%%}+%%%{%%{[%%%{-6,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[2,1,2,1,1]%%%}+%%%{%%{[%%%{12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,1,0,3,0]%%%}+%%%{%%{poly1[%%%{3,[1]%%%},0
]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,3,0,2]%%%}+%%%{%%{poly1[%%%{-12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,1,2,1]%
%%}+%%%{%%%{-6,[2]%%%},[1,2,0,3,0]%%%}+%%%{%%%{12,[2]%%%},[1,1,1,2,1]%%%}+%%%{%%%{-6,[2]%%%},[1,0,2,1,2]%%%}+%
%%{%%{[%%%{1,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,3,0,3,0]%%%}+%%%{%%{[%%%{-3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[0,2,1,2,1]%%%}+%%%{%%{[%%%{3,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,1,2,1,2]%%%}+%%%{%%{poly1[%%%{-1,[1]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3,0,3]%%%} Error: Bad Argument Value

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maple [B]  time = 0.06, size = 7299, normalized size = 30.93 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)^3,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(a*e^2-b*d*e>0)', see `assume?`
for more details)Is a*e^2-b*d*e                            +c*d^2 zero or nonzero?

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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 )}^3} \,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)^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \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)**3,x)

[Out]

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

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