3.9.32 \(\int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx\) [832]

Optimal. Leaf size=195 \[ -\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \tanh ^{-1}\left (\frac {e+d x}{\sqrt {d^2-e^2} \sqrt {-1+x^2}}\right )}{2 \left (d^2-e^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1665, 821, 739, 212} \begin {gather*} -\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac {\tanh ^{-1}\left (\frac {d x+e}{\sqrt {x^2-1} \sqrt {d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}}+\frac {\sqrt {x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((c*d^2 - b*d*e + a*e^2)*Sqrt[-1 + x^2])/(e*(d^2 - e^2)*(d + e*x)^2) + ((c*(d^3 - 4*d*e^2) - e*(3*a*d*e -
b*(d^2 + 2*e^2)))*Sqrt[-1 + x^2])/(2*e*(d^2 - e^2)^2*(d + e*x)) - ((3*b*d*e - a*(2*d^2 + e^2) - c*(d^2 + 2*e^
2))*ArcTanh[(e + d*x)/(Sqrt[d^2 - e^2]*Sqrt[-1 + x^2])])/(2*(d^2 - e^2)^(5/2))

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 739

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

Rule 821

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

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 1.02, size = 179, normalized size = 0.92 \begin {gather*} \frac {\frac {(d-e) (d+e) \sqrt {-1+x^2} \left (a e \left (-4 d^2+e^2-3 d e x\right )+c d \left (-3 d e+d^2 x-4 e^2 x\right )+b \left (2 d^3+d e^2+d^2 e x+2 e^3 x\right )\right )}{(d+e x)^2}+2 \sqrt {-d^2+e^2} \left (-3 b d e+a \left (2 d^2+e^2\right )+c \left (d^2+2 e^2\right )\right ) \tan ^{-1}\left (\frac {d+e \left (x-\sqrt {-1+x^2}\right )}{\sqrt {-d^2+e^2}}\right )}{2 (d-e)^3 (d+e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((d - e)*(d + e)*Sqrt[-1 + x^2]*(a*e*(-4*d^2 + e^2 - 3*d*e*x) + c*d*(-3*d*e + d^2*x - 4*e^2*x) + b*(2*d^3 + d
*e^2 + d^2*e*x + 2*e^3*x)))/(d + e*x)^2 + 2*Sqrt[-d^2 + e^2]*(-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*Ar
cTan[(d + e*(x - Sqrt[-1 + x^2]))/Sqrt[-d^2 + e^2]])/(2*(d - e)^3*(d + e)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(179)=358\).
time = 0.52, size = 729, normalized size = 3.74

method	result	size
default	\(-\frac {c \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}+\frac {\left (e b -2 c d \right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{\left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {d e \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {\left (a \,e^{2}-d e b +c \,d^{2}\right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{2 \left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 d e \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{\left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {d e \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (d^{2}-e^{2}\right )}+\frac {e^{2} \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{e^{5}}\)	\(729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/e^3/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e
)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))+(b*e-2*c*d)/e^4*(-1/(d^2-e^2)*e^2/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/
e^2)^(1/2)-d*e/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d
/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e)))+(a*e^2-b*d*e+c*d^2)/e^5*(-1/2/(d^2-e^2)*e^2/(x+d/e)^2*((x+
d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)+3/2*d*e/(d^2-e^2)*(-1/(d^2-e^2)*e^2/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)
+(d^2-e^2)/e^2)^(1/2)-d*e/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^
(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e)))+1/2/(d^2-e^2)*e^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(
d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^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)/(e*x+d)^3/(x^2-1)^(1/2),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(d-%e>0)', see `assume?` for mo
re details)I

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (175) = 350\).
time = 0.40, size = 1122, normalized size = 5.75 \begin {gather*} \left [\frac {c d^{7} - 2 \, b x^{2} e^{7} + {\left ({\left (2 \, a + c\right )} d^{4} e^{2} + {\left (a + 2 \, c\right )} x^{2} e^{6} - {\left (3 \, b d x^{2} - 2 \, {\left (a + 2 \, c\right )} d x\right )} e^{5} + {\left ({\left (2 \, a + c\right )} d^{2} x^{2} - 6 \, b d^{2} x + {\left (a + 2 \, c\right )} d^{2}\right )} e^{4} + {\left (2 \, {\left (2 \, a + c\right )} d^{3} x - 3 \, b d^{3}\right )} e^{3}\right )} \sqrt {d^{2} - e^{2}} \log \left (\frac {d^{2} x + d e + \sqrt {d^{2} - e^{2}} {\left (d x + e\right )} + {\left (d^{2} + \sqrt {d^{2} - e^{2}} d - e^{2}\right )} \sqrt {x^{2} - 1}}{x e + d}\right ) + {\left ({\left (3 \, a + 4 \, c\right )} d x^{2} - 4 \, b d x\right )} e^{6} + {\left (b d^{2} x^{2} + 2 \, {\left (3 \, a + 4 \, c\right )} d^{2} x - 2 \, b d^{2}\right )} e^{5} - {\left ({\left (3 \, a + 5 \, c\right )} d^{3} x^{2} - 2 \, b d^{3} x - {\left (3 \, a + 4 \, c\right )} d^{3}\right )} e^{4} + {\left (b d^{4} x^{2} - 2 \, {\left (3 \, a + 5 \, c\right )} d^{4} x + b d^{4}\right )} e^{3} + {\left (c d^{5} x^{2} + 2 \, b d^{5} x - {\left (3 \, a + 5 \, c\right )} d^{5}\right )} e^{2} + {\left (2 \, c d^{6} x + b d^{6}\right )} e - \sqrt {x^{2} - 1} {\left ({\left (2 \, b x + a\right )} e^{7} - {\left ({\left (3 \, a + 4 \, c\right )} d x - b d\right )} e^{6} - {\left (b d^{2} x + {\left (5 \, a + 3 \, c\right )} d^{2}\right )} e^{5} + {\left ({\left (3 \, a + 5 \, c\right )} d^{3} x + b d^{3}\right )} e^{4} - {\left (b d^{4} x - {\left (4 \, a + 3 \, c\right )} d^{4}\right )} e^{3} - {\left (c d^{5} x + 2 \, b d^{5}\right )} e^{2}\right )}}{2 \, {\left (2 \, d^{7} x e^{3} + d^{8} e^{2} - 6 \, d^{5} x e^{5} + 6 \, d^{3} x e^{7} - x^{2} e^{10} - 2 \, d x e^{9} + {\left (3 \, d^{2} x^{2} - d^{2}\right )} e^{8} - 3 \, {\left (d^{4} x^{2} - d^{4}\right )} e^{6} + {\left (d^{6} x^{2} - 3 \, d^{6}\right )} e^{4}\right )}}, \frac {c d^{7} - 2 \, b x^{2} e^{7} - 2 \, {\left ({\left (2 \, a + c\right )} d^{4} e^{2} + {\left (a + 2 \, c\right )} x^{2} e^{6} - {\left (3 \, b d x^{2} - 2 \, {\left (a + 2 \, c\right )} d x\right )} e^{5} + {\left ({\left (2 \, a + c\right )} d^{2} x^{2} - 6 \, b d^{2} x + {\left (a + 2 \, c\right )} d^{2}\right )} e^{4} + {\left (2 \, {\left (2 \, a + c\right )} d^{3} x - 3 \, b d^{3}\right )} e^{3}\right )} \sqrt {-d^{2} + e^{2}} \arctan \left (\frac {\sqrt {-d^{2} + e^{2}} {\left (x e - \sqrt {x^{2} - 1} e + d\right )}}{d^{2} - e^{2}}\right ) + {\left ({\left (3 \, a + 4 \, c\right )} d x^{2} - 4 \, b d x\right )} e^{6} + {\left (b d^{2} x^{2} + 2 \, {\left (3 \, a + 4 \, c\right )} d^{2} x - 2 \, b d^{2}\right )} e^{5} - {\left ({\left (3 \, a + 5 \, c\right )} d^{3} x^{2} - 2 \, b d^{3} x - {\left (3 \, a + 4 \, c\right )} d^{3}\right )} e^{4} + {\left (b d^{4} x^{2} - 2 \, {\left (3 \, a + 5 \, c\right )} d^{4} x + b d^{4}\right )} e^{3} + {\left (c d^{5} x^{2} + 2 \, b d^{5} x - {\left (3 \, a + 5 \, c\right )} d^{5}\right )} e^{2} + {\left (2 \, c d^{6} x + b d^{6}\right )} e - \sqrt {x^{2} - 1} {\left ({\left (2 \, b x + a\right )} e^{7} - {\left ({\left (3 \, a + 4 \, c\right )} d x - b d\right )} e^{6} - {\left (b d^{2} x + {\left (5 \, a + 3 \, c\right )} d^{2}\right )} e^{5} + {\left ({\left (3 \, a + 5 \, c\right )} d^{3} x + b d^{3}\right )} e^{4} - {\left (b d^{4} x - {\left (4 \, a + 3 \, c\right )} d^{4}\right )} e^{3} - {\left (c d^{5} x + 2 \, b d^{5}\right )} e^{2}\right )}}{2 \, {\left (2 \, d^{7} x e^{3} + d^{8} e^{2} - 6 \, d^{5} x e^{5} + 6 \, d^{3} x e^{7} - x^{2} e^{10} - 2 \, d x e^{9} + {\left (3 \, d^{2} x^{2} - d^{2}\right )} e^{8} - 3 \, {\left (d^{4} x^{2} - d^{4}\right )} e^{6} + {\left (d^{6} x^{2} - 3 \, d^{6}\right )} e^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(c*d^7 - 2*b*x^2*e^7 + ((2*a + c)*d^4*e^2 + (a + 2*c)*x^2*e^6 - (3*b*d*x^2 - 2*(a + 2*c)*d*x)*e^5 + ((2*a
+ c)*d^2*x^2 - 6*b*d^2*x + (a + 2*c)*d^2)*e^4 + (2*(2*a + c)*d^3*x - 3*b*d^3)*e^3)*sqrt(d^2 - e^2)*log((d^2*x
+ d*e + sqrt(d^2 - e^2)*(d*x + e) + (d^2 + sqrt(d^2 - e^2)*d - e^2)*sqrt(x^2 - 1))/(x*e + d)) + ((3*a + 4*c)*
d*x^2 - 4*b*d*x)*e^6 + (b*d^2*x^2 + 2*(3*a + 4*c)*d^2*x - 2*b*d^2)*e^5 - ((3*a + 5*c)*d^3*x^2 - 2*b*d^3*x - (3
*a + 4*c)*d^3)*e^4 + (b*d^4*x^2 - 2*(3*a + 5*c)*d^4*x + b*d^4)*e^3 + (c*d^5*x^2 + 2*b*d^5*x - (3*a + 5*c)*d^5)
*e^2 + (2*c*d^6*x + b*d^6)*e - sqrt(x^2 - 1)*((2*b*x + a)*e^7 - ((3*a + 4*c)*d*x - b*d)*e^6 - (b*d^2*x + (5*a
+ 3*c)*d^2)*e^5 + ((3*a + 5*c)*d^3*x + b*d^3)*e^4 - (b*d^4*x - (4*a + 3*c)*d^4)*e^3 - (c*d^5*x + 2*b*d^5)*e^2)
)/(2*d^7*x*e^3 + d^8*e^2 - 6*d^5*x*e^5 + 6*d^3*x*e^7 - x^2*e^10 - 2*d*x*e^9 + (3*d^2*x^2 - d^2)*e^8 - 3*(d^4*x
^2 - d^4)*e^6 + (d^6*x^2 - 3*d^6)*e^4), 1/2*(c*d^7 - 2*b*x^2*e^7 - 2*((2*a + c)*d^4*e^2 + (a + 2*c)*x^2*e^6 -
(3*b*d*x^2 - 2*(a + 2*c)*d*x)*e^5 + ((2*a + c)*d^2*x^2 - 6*b*d^2*x + (a + 2*c)*d^2)*e^4 + (2*(2*a + c)*d^3*x -
3*b*d^3)*e^3)*sqrt(-d^2 + e^2)*arctan(sqrt(-d^2 + e^2)*(x*e - sqrt(x^2 - 1)*e + d)/(d^2 - e^2)) + ((3*a + 4*c
)*d*x^2 - 4*b*d*x)*e^6 + (b*d^2*x^2 + 2*(3*a + 4*c)*d^2*x - 2*b*d^2)*e^5 - ((3*a + 5*c)*d^3*x^2 - 2*b*d^3*x -
(3*a + 4*c)*d^3)*e^4 + (b*d^4*x^2 - 2*(3*a + 5*c)*d^4*x + b*d^4)*e^3 + (c*d^5*x^2 + 2*b*d^5*x - (3*a + 5*c)*d^
5)*e^2 + (2*c*d^6*x + b*d^6)*e - sqrt(x^2 - 1)*((2*b*x + a)*e^7 - ((3*a + 4*c)*d*x - b*d)*e^6 - (b*d^2*x + (5*
a + 3*c)*d^2)*e^5 + ((3*a + 5*c)*d^3*x + b*d^3)*e^4 - (b*d^4*x - (4*a + 3*c)*d^4)*e^3 - (c*d^5*x + 2*b*d^5)*e^
2))/(2*d^7*x*e^3 + d^8*e^2 - 6*d^5*x*e^5 + 6*d^3*x*e^7 - x^2*e^10 - 2*d*x*e^9 + (3*d^2*x^2 - d^2)*e^8 - 3*(d^4
*x^2 - d^4)*e^6 + (d^6*x^2 - 3*d^6)*e^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x + c x^{2}}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \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)/(e*x+d)**3/(x**2-1)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (175) = 350\).
time = 1.74, size = 536, normalized size = 2.75 \begin {gather*} \frac {{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (-\frac {{\left (x - \sqrt {x^{2} - 1}\right )} e + d}{\sqrt {-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt {-d^{2} + e^{2}}} + \frac {2 \, c d^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e + 2 \, c d^{5} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, b d^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e - 2 \, a d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{3} - 5 \, c d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{3} - 6 \, a d^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{2} - 7 \, c d^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{2} + 2 \, c d^{4} {\left (x - \sqrt {x^{2} - 1}\right )} e + 3 \, b d {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{4} + 5 \, b d^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{3} + 4 \, b d^{3} {\left (x - \sqrt {x^{2} - 1}\right )} e^{2} - a {\left (x - \sqrt {x^{2} - 1}\right )}^{3} e^{5} - 3 \, a d {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{4} - 4 \, c d {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{4} - 10 \, a d^{2} {\left (x - \sqrt {x^{2} - 1}\right )} e^{3} - 11 \, c d^{2} {\left (x - \sqrt {x^{2} - 1}\right )} e^{3} + c d^{3} e^{2} + 2 \, b {\left (x - \sqrt {x^{2} - 1}\right )}^{2} e^{5} + 5 \, b d {\left (x - \sqrt {x^{2} - 1}\right )} e^{4} + b d^{2} e^{3} + a {\left (x - \sqrt {x^{2} - 1}\right )} e^{5} - 3 \, a d e^{4} - 4 \, c d e^{4} + 2 \, b e^{5}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )} {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} e + 2 \, d {\left (x - \sqrt {x^{2} - 1}\right )} + e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(-((x - sqrt(x^2 - 1))*e + d)/sqrt(-d^2 + e^2))/((d^4 - 2*
d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) + (2*c*d^4*(x - sqrt(x^2 - 1))^3*e + 2*c*d^5*(x - sqrt(x^2 - 1))^2 + 2*b*d^4*
(x - sqrt(x^2 - 1))^2*e - 2*a*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 5*c*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 6*a*d^3*(x -
sqrt(x^2 - 1))^2*e^2 - 7*c*d^3*(x - sqrt(x^2 - 1))^2*e^2 + 2*c*d^4*(x - sqrt(x^2 - 1))*e + 3*b*d*(x - sqrt(x^
2 - 1))^3*e^4 + 5*b*d^2*(x - sqrt(x^2 - 1))^2*e^3 + 4*b*d^3*(x - sqrt(x^2 - 1))*e^2 - a*(x - sqrt(x^2 - 1))^3*
e^5 - 3*a*d*(x - sqrt(x^2 - 1))^2*e^4 - 4*c*d*(x - sqrt(x^2 - 1))^2*e^4 - 10*a*d^2*(x - sqrt(x^2 - 1))*e^3 - 1
1*c*d^2*(x - sqrt(x^2 - 1))*e^3 + c*d^3*e^2 + 2*b*(x - sqrt(x^2 - 1))^2*e^5 + 5*b*d*(x - sqrt(x^2 - 1))*e^4 +
b*d^2*e^3 + a*(x - sqrt(x^2 - 1))*e^5 - 3*a*d*e^4 - 4*c*d*e^4 + 2*b*e^5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*((x - sq
rt(x^2 - 1))^2*e + 2*d*(x - sqrt(x^2 - 1)) + e)^2)

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

[Out]

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

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