3.9.82 \(\int \frac {1}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [882]

Optimal. Leaf size=155 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {e^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

e^2*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)^
(3/2)-2*(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)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {754, 12, 738, 212} \begin {gather*} \frac {e^2 \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 )^{3/2}}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.15, size = 187, normalized size = 1.21 \begin {gather*} -\frac {2 \left (\left (c d^2+e (-b d+a e)\right ) \left (-b^2 e+2 c (a e+c d x)+b c (d-e x)\right )+\left (-b^2+4 a c\right ) e^2 \sqrt {-c d^2+b d e-a e^2} \sqrt {a+x (b+c x)} \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(145)=290\).
time = 0.11, size = 400, normalized size = 2.58

method	result	size
default	\(\frac {\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (e b -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {e b -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (e b -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{e}\)	\(400\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(1/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e
/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+
(b*e-2*c*d)/e*(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+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d
)/e*(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)^(3/2),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. 631 vs. \(2 (149) = 298\).
time = 6.99, size = 1306, normalized size = 8.43 \begin {gather*} \left [\frac {{\left (a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt {c d^{2} - b d e + a e^{2}} e^{2} \log \left (-\frac {8 \, c^{2} d^{2} x^{2} + 8 \, b c d^{2} x + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c d^{2} - b d e + a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a} + {\left (8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{2} + 4 \, a b d + {\left (3 \, b^{2} + 4 \, a c\right )} d x\right )} e}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 4 \, {\left (2 \, c^{3} d^{3} x + b c^{2} d^{3} - {\left (a b c x + a b^{2} - 2 \, a^{2} c\right )} e^{3} + {\left ({\left (b^{2} c + 2 \, a c^{2}\right )} d x + {\left (b^{3} - a b c\right )} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, {\left (b^{2} c - a c^{2}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{2} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{2} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e\right )}}, \frac {{\left (a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2} + {\left (a c x^{2} + a b x + a^{2}\right )} e^{2} - {\left (b c d x^{2} + b^{2} d x + a b d\right )} e\right )}}\right ) e^{2} - 2 \, {\left (2 \, c^{3} d^{3} x + b c^{2} d^{3} - {\left (a b c x + a b^{2} - 2 \, a^{2} c\right )} e^{3} + {\left ({\left (b^{2} c + 2 \, a c^{2}\right )} d x + {\left (b^{3} - a b c\right )} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, {\left (b^{2} c - a c^{2}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} + {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} e^{4} - 2 \, {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{2} + {\left (a b^{4} - 4 \, a^{2} b^{2} c\right )} d x + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d\right )} e^{3} + {\left ({\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{2} x^{2} + {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} d^{2} x + {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2}\right )} e^{2} - 2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{3} x^{2} + {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} d^{3} x + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{3}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/2*((a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*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)*sq
rt(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*(2*c^3*d^3*x + b*c^2*d^3 - (a*b*c*x + a*b^2 - 2*a^2*c)*e^3 + ((b^2*c + 2
*a*c^2)*d*x + (b^3 - a*b*c)*d)*e^2 - (3*b*c^2*d^2*x + 2*(b^2*c - a*c^2)*d^2)*e)*sqrt(c*x^2 + b*x + a))/((b^2*c
^3 - 4*a*c^4)*d^4*x^2 + (b^3*c^2 - 4*a*b*c^3)*d^4*x + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*
b^2*c - 4*a^3*c^2)*x^2 + (a^2*b^3 - 4*a^3*b*c)*x)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^2 + (a*b^4 - 4*a^2*b^2*
c)*d*x + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^2 + (b^5 - 2*a*b^3*c - 8*a^2*
b*c^2)*d^2*x + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^2 + (b^4*c - 4*a*b^
2*c^2)*d^3*x + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e), ((a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*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^2 - 2*(2*c^3*d^3*x + b*c^2*d^3 - (a*b*c*x + a*b^2 - 2*a^2*c)*e^3 + ((b^2*c + 2*a*c^2)*d*x + (b^3 - a*b
*c)*d)*e^2 - (3*b*c^2*d^2*x + 2*(b^2*c - a*c^2)*d^2)*e)*sqrt(c*x^2 + b*x + a))/((b^2*c^3 - 4*a*c^4)*d^4*x^2 +
(b^3*c^2 - 4*a*b*c^3)*d^4*x + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^2 +
(a^2*b^3 - 4*a^3*b*c)*x)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^2 + (a*b^4 - 4*a^2*b^2*c)*d*x + (a^2*b^3 - 4*a^
3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^2 + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x + (a*b^4 -
2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^2 + (b^4*c - 4*a*b^2*c^2)*d^3*x + (a*b^3*c
- 4*a^2*b*c^2)*d^3)*e)]

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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 {3}{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)**(3/2),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(3/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (149) = 298\).
time = 4.35, size = 447, normalized size = 2.88 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2} + 2 \, a c^{2} d e^{2} - a b c e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + 2 \, a c^{2} d^{2} e + b^{3} d e^{2} - a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right ) e^{2}}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,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)^(3/2)),x)

[Out]

int(1/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x)

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