3.24.97 \(\int \frac {1}{(d+e x) (a+b x+c x^2)^{5/2}} \, dx\) [2397]

Optimal. Leaf size=310 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {e^4 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {754, 836, 12, 738, 212} \begin {gather*} -\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-4 c e (b d-3 a e)-3 b^2 e^2+8 c^2 d^2\right )+4 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e^4 \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 )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 836

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )+2 c e (2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {3 \left (b^2-4 a c\right )^2 e^4}{4 (d+e x) \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {e^4 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (2 e^4\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 )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-3 b^2 e^2-4 c e (2 b d-5 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {e^4 \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 )}{\left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 10.32, size = 309, normalized size = 1.00 \begin {gather*} -\frac {2 \left (-b^2 e+2 c (a e+c d x)+b c (d-e x)\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}+\frac {2 \left (3 b^4 e^3+b^3 c e^2 (d+3 e x)+8 c^2 \left (3 a^2 e^3+2 c^2 d^3 x+5 a c d e^2 x\right )+4 b c^2 \left (2 c d^2 (d-3 e x)+5 a e^2 (d-e x)\right )+2 b^2 c e \left (-11 a e^2+c d (-6 d+e x)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}-\frac {e^4 \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 )}{\left (c d^2+e (-b d+a e)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x)))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*
x))^(3/2)) + (2*(3*b^4*e^3 + b^3*c*e^2*(d + 3*e*x) + 8*c^2*(3*a^2*e^3 + 2*c^2*d^3*x + 5*a*c*d*e^2*x) + 4*b*c^2
*(2*c*d^2*(d - 3*e*x) + 5*a*e^2*(d - e*x)) + 2*b^2*c*e*(-11*a*e^2 + c*d*(-6*d + e*x))))/(3*(b^2 - 4*a*c)^2*(c*
d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) - (e^4*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)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(294)=588\).
time = 0.91, size = 766, normalized size = 2.47

method	result	size
default	\(\frac {\frac {e^{2}}{3 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}-\frac {e \left (b e -2 c d \right ) \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a -b d e +c \,d^{2}}}{e}\)	\(766\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(1/3/(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)^(3/2)-1/2*e*(b*
e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)
^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)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/
e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(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/(a*e^2-b*d*e+c*d^2)*e^2*(1/(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)-e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(4*c*(a*e^2-b*d*e+c*d^2)/
e^2-1/e^2*(b*e-2*c*d)^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/(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(1/(e*x+d)/(c*x^2+b*x+a)^(5/2),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. 2253 vs. \(2 (305) = 610\).
time = 7.61, size = 4549, normalized size = 14.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^
2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt(c*
d^2 - b*d*e + a*e^2)*e^4*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*(16*c^6*d^5*x^3 + 24*b*c^5*d^5*x^2 +
6*(b^2*c^4 + 4*a*c^5)*d^5*x - (b^3*c^3 - 12*a*b*c^4)*d^5 + (4*a^2*b^4 - 28*a^3*b^2*c + 32*a^4*c^2 + (3*a*b^3*c
^2 - 20*a^2*b*c^3)*x^3 + 6*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*x^2 + 3*(a*b^5 - 6*a^2*b^3*c)*x)*e^5 - ((3*b^
4*c^2 - 22*a*b^2*c^3 - 40*a^2*c^4)*d*x^3 + 3*(2*b^5*c - 15*a*b^3*c^2 - 12*a^2*b*c^3)*d*x^2 + 3*(b^6 - 6*a*b^4*
c - 8*a^2*b^2*c^2 - 16*a^3*c^3)*d*x + (5*a*b^5 - 34*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^4 + ((b^3*c^3 - 84*a*b*c^4)
*d^2*x^3 + 3*(b^4*c^2 - 46*a*b^2*c^3 + 8*a^2*c^4)*d^2*x^2 + 3*(b^5*c - 17*a*b^3*c^2 - 28*a^2*b*c^3)*d^2*x + (b
^6 - 66*a^2*b^2*c^2 + 40*a^3*c^3)*d^2)*e^3 + (2*(13*b^2*c^4 + 28*a*c^5)*d^3*x^3 + 3*(13*b^3*c^3 + 28*a*b*c^4)*
d^3*x^2 + 3*(3*b^4*c^2 + 22*a*b^2*c^3 + 24*a^2*c^4)*d^3*x - (3*b^5*c - 27*a*b^3*c^2 - 20*a^2*b*c^3)*d^3)*e^2 -
(40*b*c^5*d^4*x^3 + 60*b^2*c^4*d^4*x^2 + 15*(b^3*c^3 + 4*a*b*c^4)*d^4*x - (3*b^4*c^2 - 34*a*b^2*c^3 + 8*a^2*c
^4)*d^4)*e)*sqrt(c*x^2 + b*x + a))/((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 +
16*a^2*b*c^6)*d^6*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*d^6*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b
*c^5)*d^6*x + (a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*
c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^
4*c + 32*a^6*c^3)*x^2 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x)*e^6 - 3*((a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16
*a^4*b*c^4)*d*x^4 + 2*(a^2*b^6*c - 8*a^3*b^4*c^2 + 16*a^4*b^2*c^3)*d*x^3 + (a^2*b^7 - 6*a^3*b^5*c + 32*a^5*b*c
^3)*d*x^2 + 2*(a^3*b^6 - 8*a^4*b^4*c + 16*a^5*b^2*c^2)*d*x + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d)*e^5 + 3
*((a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*x^4 + 2*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^
3 + 16*a^4*b*c^4)*d^2*x^3 + (a*b^8 - 5*a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^4*b^2*c^3 + 32*a^5*c^4)*d^2*x^2 + 2*(a
^2*b^7 - 7*a^3*b^5*c + 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2*x + (a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c
^3)*d^2)*e^4 - ((b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*x^4 + 2*(b^8*c - 2*a*b^6*c^2 - 32*
a^2*b^4*c^3 + 96*a^3*b^2*c^4)*d^3*x^3 + (b^9 - 36*a^2*b^5*c^2 + 32*a^3*b^3*c^3 + 192*a^4*b*c^4)*d^3*x^2 + 2*(a
*b^8 - 2*a^2*b^6*c - 32*a^3*b^4*c^2 + 96*a^4*b^2*c^3)*d^3*x + (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5
*b*c^3)*d^3)*e^3 + 3*((b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*x^4 + 2*(b^7*c^2 - 7*a*b^5*c^3
+ 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^4*x^3 + (b^8*c - 5*a*b^6*c^2 - 6*a^2*b^4*c^3 + 32*a^3*b^2*c^4 + 32*a^4*c^5)*
d^4*x^2 + 2*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^4*x + (a^2*b^6*c - 7*a^3*b^4*c^2 + 8*a^
4*b^2*c^3 + 16*a^5*c^4)*d^4)*e^2 - 3*((b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*x^4 + 2*(b^6*c^3 - 8*a*b^4*c^
4 + 16*a^2*b^2*c^5)*d^5*x^3 + (b^7*c^2 - 6*a*b^5*c^3 + 32*a^3*b*c^5)*d^5*x^2 + 2*(a*b^6*c^2 - 8*a^2*b^4*c^3 +
16*a^3*b^2*c^4)*d^5*x + (a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5)*e), 1/3*(3*(a^2*b^4 - 8*a^3*b^2*c +
16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*
a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*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))*e^4 + 2*(16*c^6*d^5*x^3 + 24*b*c^5*
d^5*x^2 + 6*(b^2*c^4 + 4*a*c^5)*d^5*x - (b^3*c^3 - 12*a*b*c^4)*d^5 + (4*a^2*b^4 - 28*a^3*b^2*c + 32*a^4*c^2 +
(3*a*b^3*c^2 - 20*a^2*b*c^3)*x^3 + 6*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*x^2 + 3*(a*b^5 - 6*a^2*b^3*c)*x)*e^
5 - ((3*b^4*c^2 - 22*a*b^2*c^3 - 40*a^2*c^4)*d*x^3 + 3*(2*b^5*c - 15*a*b^3*c^2 - 12*a^2*b*c^3)*d*x^2 + 3*(b^6
- 6*a*b^4*c - 8*a^2*b^2*c^2 - 16*a^3*c^3)*d*x + (5*a*b^5 - 34*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^4 + ((b^3*c^3 - 8
4*a*b*c^4)*d^2*x^3 + 3*(b^4*c^2 - 46*a*b^2*c^3 + 8*a^2*c^4)*d^2*x^2 + 3*(b^5*c - 17*a*b^3*c^2 - 28*a^2*b*c^3)*
d^2*x + (b^6 - 66*a^2*b^2*c^2 + 40*a^3*c^3)*d^2)*e^3 + (2*(13*b^2*c^4 + 28*a*c^5)*d^3*x^3 + 3*(13*b^3*c^3 + 28
*a*b*c^4)*d^3*x^2 + 3*(3*b^4*c^2 + 22*a*b^2*c^3 + 24*a^2*c^4)*d^3*x - (3*b^5*c - 27*a*b^3*c^2 - 20*a^2*b*c^3)*
d^3)*e^2 - (40*b*c^5*d^4*x^3 + 60*b^2*c^4*d^4*x^2 + 15*(b^3*c^3 + 4*a*b*c^4)*d^4*x - (3*b^4*c^2 - 34*a*b^2*c^3
+ 8*a^2*c^4)*d^4)*e)*sqrt(c*x^2 + b*x + a))/((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*x^4 + 2*(b^5*c^4 - 8*a*
b^3*c^5 + 16*a^2*b*c^6)*d^6*x^3 + (b^6*c^3 - 6*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(5/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7942 vs. \(2 (305) = 610\).
time = 2.14, size = 7942, normalized size = 25.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

2*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))*e^4/((c^2*d^4 - 2*
b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 2/3*((((16*c^
11*d^15 - 120*b*c^10*d^14*e + 386*b^2*c^9*d^13*e^2 + 136*a*c^10*d^13*e^2 - 689*b^3*c^8*d^12*e^3 - 884*a*b*c^9*
d^12*e^3 + 732*b^4*c^7*d^11*e^4 + 2412*a*b^2*c^8*d^11*e^4 + 480*a^2*c^9*d^11*e^4 - 451*b^5*c^6*d^10*e^5 - 3542
*a*b^3*c^7*d^10*e^5 - 2640*a^2*b*c^8*d^10*e^5 + 130*b^6*c^5*d^9*e^6 + 2950*a*b^4*c^6*d^9*e^6 + 5910*a^2*b^2*c^
7*d^9*e^6 + 920*a^3*c^8*d^9*e^6 + 9*b^7*c^4*d^8*e^7 - 1296*a*b^5*c^5*d^8*e^7 - 6795*a^2*b^3*c^6*d^8*e^7 - 4140
*a^3*b*c^7*d^8*e^7 - 16*b^8*c^3*d^7*e^8 + 184*a*b^6*c^4*d^7*e^8 + 4080*a^2*b^4*c^5*d^7*e^8 + 7240*a^3*b^2*c^6*
d^7*e^8 + 1040*a^4*c^7*d^7*e^8 + 3*b^9*c^2*d^6*e^9 + 58*a*b^7*c^3*d^6*e^9 - 1050*a^2*b^5*c^4*d^6*e^9 - 6020*a^
3*b^3*c^5*d^6*e^9 - 3640*a^4*b*c^6*d^6*e^9 - 18*a*b^8*c^2*d^5*e^10 - 30*a^2*b^6*c^3*d^5*e^10 + 2220*a^3*b^4*c^
4*d^5*e^10 + 4590*a^4*b^2*c^5*d^5*e^10 + 696*a^5*c^6*d^5*e^10 + 45*a^2*b^7*c^2*d^4*e^11 - 160*a^3*b^5*c^3*d^4*
e^11 - 2375*a^4*b^3*c^4*d^4*e^11 - 1740*a^5*b*c^5*d^4*e^11 - 60*a^3*b^6*c^2*d^3*e^12 + 340*a^4*b^4*c^3*d^3*e^1
2 + 1356*a^5*b^2*c^4*d^3*e^12 + 256*a^6*c^5*d^3*e^12 + 45*a^4*b^5*c^2*d^2*e^13 - 294*a^5*b^3*c^3*d^2*e^13 - 38
4*a^6*b*c^4*d^2*e^13 - 18*a^5*b^4*c^2*d*e^14 + 122*a^6*b^2*c^3*d*e^14 + 40*a^7*c^4*d*e^14 + 3*a^6*b^3*c^2*e^15
- 20*a^7*b*c^3*e^15)*x/(b^4*c^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^15*e + 64*a*b^3*c^8*
d^15*e - 128*a^2*b*c^9*d^15*e + 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2*c^8*d^14*e^2 + 128*
a^3*c^9*d^14*e^2 - 56*b^7*c^5*d^13*e^3 + 392*a*b^5*c^6*d^13*e^3 - 448*a^2*b^3*c^7*d^13*e^3 - 896*a^3*b*c^8*d^1
3*e^3 + 70*b^8*c^4*d^12*e^4 - 392*a*b^6*c^5*d^12*e^4 - 196*a^2*b^4*c^6*d^12*e^4 + 2464*a^3*b^2*c^7*d^12*e^4 +
448*a^4*c^8*d^12*e^4 - 56*b^9*c^3*d^11*e^5 + 168*a*b^7*c^4*d^11*e^5 + 1176*a^2*b^5*c^5*d^11*e^5 - 3136*a^3*b^3
*c^6*d^11*e^5 - 2688*a^4*b*c^7*d^11*e^5 + 28*b^10*c^2*d^10*e^6 + 56*a*b^8*c^3*d^10*e^6 - 1372*a^2*b^6*c^4*d^10
*e^6 + 1176*a^3*b^4*c^5*d^10*e^6 + 6272*a^4*b^2*c^6*d^10*e^6 + 896*a^5*c^7*d^10*e^6 - 8*b^11*c*d^9*e^7 - 104*a
*b^9*c^2*d^9*e^7 + 656*a^2*b^7*c^3*d^9*e^7 + 1512*a^3*b^5*c^4*d^9*e^7 - 6720*a^4*b^3*c^5*d^9*e^7 - 4480*a^5*b*
c^6*d^9*e^7 + b^12*d^8*e^8 + 48*a*b^10*c*d^8*e^8 - 12*a^2*b^8*c^2*d^8*e^8 - 1904*a^3*b^6*c^3*d^8*e^8 + 2310*a^
4*b^4*c^4*d^8*e^8 + 8400*a^5*b^2*c^5*d^8*e^8 + 1120*a^6*c^6*d^8*e^8 - 8*a*b^11*d^7*e^9 - 104*a^2*b^9*c*d^7*e^9
+ 656*a^3*b^7*c^2*d^7*e^9 + 1512*a^4*b^5*c^3*d^7*e^9 - 6720*a^5*b^3*c^4*d^7*e^9 - 4480*a^6*b*c^5*d^7*e^9 + 28
*a^2*b^10*d^6*e^10 + 56*a^3*b^8*c*d^6*e^10 - 1372*a^4*b^6*c^2*d^6*e^10 + 1176*a^5*b^4*c^3*d^6*e^10 + 6272*a^6*
b^2*c^4*d^6*e^10 + 896*a^7*c^5*d^6*e^10 - 56*a^3*b^9*d^5*e^11 + 168*a^4*b^7*c*d^5*e^11 + 1176*a^5*b^5*c^2*d^5*
e^11 - 3136*a^6*b^3*c^3*d^5*e^11 - 2688*a^7*b*c^4*d^5*e^11 + 70*a^4*b^8*d^4*e^12 - 392*a^5*b^6*c*d^4*e^12 - 19
6*a^6*b^4*c^2*d^4*e^12 + 2464*a^7*b^2*c^3*d^4*e^12 + 448*a^8*c^4*d^4*e^12 - 56*a^5*b^7*d^3*e^13 + 392*a^6*b^5*
c*d^3*e^13 - 448*a^7*b^3*c^2*d^3*e^13 - 896*a^8*b*c^3*d^3*e^13 + 28*a^6*b^6*d^2*e^14 - 216*a^7*b^4*c*d^2*e^14
+ 384*a^8*b^2*c^2*d^2*e^14 + 128*a^9*c^3*d^2*e^14 - 8*a^7*b^5*d*e^15 + 64*a^8*b^3*c*d*e^15 - 128*a^9*b*c^2*d*e
^15 + a^8*b^4*e^16 - 8*a^9*b^2*c*e^16 + 16*a^10*c^2*e^16) + 3*(8*b*c^10*d^15 - 60*b^2*c^9*d^14*e + 193*b^3*c^8
*d^13*e^2 + 68*a*b*c^9*d^13*e^2 - 344*b^4*c^7*d^12*e^3 - 446*a*b^2*c^8*d^12*e^3 + 8*a^2*c^9*d^12*e^3 + 363*b^5
*c^6*d^11*e^4 + 1230*a*b^3*c^7*d^11*e^4 + 192*a^2*b*c^8*d^11*e^4 - 218*b^6*c^5*d^10*e^5 - 1828*a*b^4*c^6*d^10*
e^5 - 1224*a^2*b^2*c^7*d^10*e^5 + 48*a^3*c^8*d^10*e^5 + 55*b^7*c^4*d^9*e^6 + 1540*a*b^5*c^5*d^9*e^6 + 2915*a^2
*b^3*c^6*d^9*e^6 + 220*a^3*b*c^7*d^9*e^6 + 12*b^8*c^3*d^8*e^7 - 678*a*b^6*c^4*d^8*e^7 - 3510*a^2*b^4*c^5*d^8*e
^7 - 1650*a^3*b^2*c^6*d^8*e^7 + 120*a^4*c^7*d^8*e^7 - 11*b^9*c^2*d^7*e^8 + 86*a*b^7*c^3*d^7*e^8 + 2202*a^2*b^5
*c^4*d^7*e^8 + 3380*a^3*b^3*c^5*d^7*e^8 + 40*a^4*b*c^6*d^7*e^8 + 2*b^10*c*d^6*e^9 + 40*a*b^8*c^2*d^6*e^9 - 592
*a^2*b^6*c^3*d^6*e^9 - 3120*a^3*b^4*c^4*d^6*e^9 - 1180*a^4*b^2*c^5*d^6*e^9 + 160*a^5*c^6*d^6*e^9 - 12*a*b^9*c*
d^5*e^10 - 21*a^2*b^7*c^2*d^5*e^10 + 1272*a^3*b^5*c^3*d^5*e^10 + 2055*a^4*b^3*c^4*d^5*e^10 - 132*a^5*b*c^5*d^5
*e^10 + 30*a^2*b^8*c*d^4*e^11 - 110*a^3*b^6*c^2*d^4*e^11 - 1300*a^4*b^4*c^3*d^4*e^11 - 450*a^5*b^2*c^4*d^4*e^1
1 + 120*a^6*c^5*d^4*e^11 - 40*a^3*b^7*c*d^3*e^12 + 235*a^4*b^5*c^2*d^3*e^12 + 638*a^5*b^3*c^3*d^3*e^12 - 112*a
^6*b*c^4*d^3*e^12 + 30*a^4*b^6*c*d^2*e^13 - 204*a^5*b^4*c^2*d^2*e^13 - 96*a^6*b^2*c^3*d^2*e^13 + 48*a^7*c^4*d^
2*e^13 - 12*a^5*b^5*c*d*e^14 + 85*a^6*b^3*c^2*d*e^14 - 28*a^7*b*c^3*d*e^14 + 2*a^6*b^4*c*e^15 - 14*a^7*b^2*c^2
*e^15 + 8*a^8*c^3*e^15)/(b^4*c^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^15*e + 64*a*b^3*c^8*
d^15*e - 128*a^2*b*c^9*d^15*e + 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2*c^8*d^14*e^2 + 128*
a^3*c^9*d^14*e^2 - 56*b^7*c^5*d^13*e^3 + 392*a*...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)*(a + b*x + c*x^2)^(5/2)), x)

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