3.24.49 \(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [2349]

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

[Out]

-1/2*(c*x^2+b*x+a)^(3/2)/e/(e*x+d)^2-3/2*(-b*e+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/
2)/e^4+3/8*(8*c^2*d^2+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))/e^4/(a*e^2-b*d*e+c*d^2)^(1/2)+3/4*(2*c*e*x-b*e+4*c*d)*(c*x^2+b*x+a)^(1/2)/e^3/(e*x+d
)

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Rubi [A]
time = 0.16, 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 = {746, 826, 857, 635, 212, 738} \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 {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} \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 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 635

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 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 746

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 826

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 857

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)) \text {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 ) \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 )}{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 = 10.64, size = 311, normalized size = 1.32 \begin {gather*} \frac {-\frac {2 (a+x (b+c x))^{3/2}}{(d+e x)^2}+\frac {3 (2 c d-b e) (a+x (b+c x))^{3/2}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 \left (-\frac {2 e \sqrt {a+x (b+c x)} \left (-b^2 e^2+2 c^2 d (-2 d+e x)-c e (-5 b d+2 a e+b e x)\right )}{c d^2+e (-b d+a e)}-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 )-\frac {\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 x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d^2+e (-b d+a e)}}\right )}{2 e^3}}{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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1854\) vs. \(2(208)=416\).
time = 0.98, size = 1855, normalized size = 7.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1855\)
risch	\(\text {Expression too large to display}\)	\(6114\)

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,method=_RETURNVERBOSE)

[Out]

1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5
/2)+1/4*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+3/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(
b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2
)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
))+(a*e^2-b*d*e+c*d^2)/e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c
*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b*e-2*c
*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2))/(x+d/e))))+4*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d
/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d
*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*
d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))))+3/2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e
*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3
/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)))+(a*e^2-b*d*e+c*d^2)/e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2
*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b*e-2
*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2))/(x+d/e)))))

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (212) = 424\).
time = 51.53, size = 2645, normalized size = 11.21 \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)^3,x, algorithm="fricas")

[Out]

[-1/16*(12*(2*c^2*d^5 - a*b*x^2*e^5 - (2*a*b*d*x - (b^2 + 2*a*c)*d*x^2)*e^4 - (3*b*c*d^2*x^2 + a*b*d^2 - 2*(b^
2 + 2*a*c)*d^2*x)*e^3 + (2*c^2*d^3*x^2 - 6*b*c*d^3*x + (b^2 + 2*a*c)*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*s
qrt(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 +
(b^2 + 4*a*c)*x^2*e^4 - 2*(4*b*c*d*x^2 - (b^2 + 4*a*c)*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + (b^2 + 4*a*c
)*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*d^3)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2
+ 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x
+ (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^
2)) - 4*(12*c^2*d^4*e + (4*a*c*x^2 - 5*a*b*x - 2*a^2)*e^5 - (4*b*c*d*x^2 + a*b*d - (5*b^2 + 18*a*c)*d*x)*e^4 +
(4*c^2*d^2*x^2 - 23*b*c*d^2*x + (3*b^2 + 10*a*c)*d^2)*e^3 + 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x
+ a))/(c*d^4*e^4 + a*x^2*e^8 - (b*d*x^2 - 2*a*d*x)*e^7 + (c*d^2*x^2 - 2*b*d^2*x + a*d^2)*e^6 + (2*c*d^3*x - b
*d^3)*e^5), 1/16*(24*(2*c^2*d^5 - a*b*x^2*e^5 - (2*a*b*d*x - (b^2 + 2*a*c)*d*x^2)*e^4 - (3*b*c*d^2*x^2 + a*b*d
^2 - 2*(b^2 + 2*a*c)*d^2*x)*e^3 + (2*c^2*d^3*x^2 - 6*b*c*d^3*x + (b^2 + 2*a*c)*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c
*d^4)*e)*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 + (b^2 + 4*a*c)*x^2*e^4 - 2*(4*b*c*d*x^2 - (b^2 + 4*a*c)*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + (b^2 + 4
*a*c)*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*d^3)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x +
(b^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a
*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e
+ d^2)) + 4*(12*c^2*d^4*e + (4*a*c*x^2 - 5*a*b*x - 2*a^2)*e^5 - (4*b*c*d*x^2 + a*b*d - (5*b^2 + 18*a*c)*d*x)*e
^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + (3*b^2 + 10*a*c)*d^2)*e^3 + 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 +
b*x + a))/(c*d^4*e^4 + a*x^2*e^8 - (b*d*x^2 - 2*a*d*x)*e^7 + (c*d^2*x^2 - 2*b*d^2*x + a*d^2)*e^6 + (2*c*d^3*x
- b*d^3)*e^5), 1/8*(3*(8*c^2*d^4 + (b^2 + 4*a*c)*x^2*e^4 - 2*(4*b*c*d*x^2 - (b^2 + 4*a*c)*d*x)*e^3 + (8*c^2*d
^2*x^2 - 16*b*c*d^2*x + (b^2 + 4*a*c)*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*d^3)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arc
tan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c
*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) - 6*(2*c^2*d^5 - a*b*x^2*e^
5 - (2*a*b*d*x - (b^2 + 2*a*c)*d*x^2)*e^4 - (3*b*c*d^2*x^2 + a*b*d^2 - 2*(b^2 + 2*a*c)*d^2*x)*e^3 + (2*c^2*d^3
*x^2 - 6*b*c*d^3*x + (b^2 + 2*a*c)*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*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 + (4*a*c*x^2 - 5*a*b*x - 2*a^2)*e
^5 - (4*b*c*d*x^2 + a*b*d - (5*b^2 + 18*a*c)*d*x)*e^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + (3*b^2 + 10*a*c)*d^2)*
e^3 + 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x + a))/(c*d^4*e^4 + a*x^2*e^8 - (b*d*x^2 - 2*a*d*x)*e^7
+ (c*d^2*x^2 - 2*b*d^2*x + a*d^2)*e^6 + (2*c*d^3*x - b*d^3)*e^5), 1/8*(3*(8*c^2*d^4 + (b^2 + 4*a*c)*x^2*e^4 -
2*(4*b*c*d*x^2 - (b^2 + 4*a*c)*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + (b^2 + 4*a*c)*d^2)*e^2 + 8*(2*c^2*d
^3*x - b*c*d^3)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x
+ 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2
+ b^2*d*x + a*b*d)*e)) + 12*(2*c^2*d^5 - a*b*x^2*e^5 - (2*a*b*d*x - (b^2 + 2*a*c)*d*x^2)*e^4 - (3*b*c*d^2*x^2
+ a*b*d^2 - 2*(b^2 + 2*a*c)*d^2*x)*e^3 + (2*c^2*d^3*x^2 - 6*b*c*d^3*x + (b^2 + 2*a*c)*d^3)*e^2 + (4*c^2*d^4*x
- 3*b*c*d^4)*e)*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 + (4*a*c*x^2 - 5*a*b*x - 2*a^2)*e^5 - (4*b*c*d*x^2 + a*b*d - (5*b^2 + 18*a*c)*d*x)*e^4 + (4*c^2*d
^2*x^2 - 23*b*c*d^2*x + (3*b^2 + 10*a*c)*d^2)*e^3 + 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x + a))/(c
*d^4*e^4 + a*x^2*e^8 - (b*d*x^2 - 2*a*d*x)*e^7 + (c*d^2*x^2 - 2*b*d^2*x + a*d^2)*e^6 + (2*c*d^3*x - b*d^3)*e^5
)]

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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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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,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[6,0,7,0,0]%%%}+%%%{%%{[-6,0]:[1,0,%%%{-1,[1]%%%}]%%},
[5,0,6,1,0]

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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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