3.2305 \(\int \frac{a+b x}{(1+x)^{5/2} (1-x+x^2)^{5/2}} \, dx\)

Optimal. Leaf size=351 \[ \frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (7 a+5 \left (1-\sqrt{3}\right ) b\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{2 x (a+b x)}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}+\frac{2 x (7 a+5 b x)}{27 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{10 b \left (x^3+1\right )}{27 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}}+\frac{5 \sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}} \]

[Out]

(2*x*(7*a + 5*b*x))/(27*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (2*x*(a + b*x))/(9*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 +
x^3)) - (10*b*(1 + x^3))/(27*Sqrt[1 + x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) + (5*Sqrt[2 - Sqrt[3]]*b*Sqrt[1
+ x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sq
rt[3]])/(9*3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2]) + (2*Sqrt[2 + Sqrt[3]]*(7*a + 5*(1 - S
qrt[3])*b)*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3]
+ x)], -7 - 4*Sqrt[3]])/(27*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

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Rubi [A]  time = 0.190158, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {809, 1855, 1878, 218, 1877} \[ \frac{2 x (a+b x)}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}+\frac{2 x (7 a+5 b x)}{27 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (7 a+5 \left (1-\sqrt{3}\right ) b\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{10 b \left (x^3+1\right )}{27 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}}+\frac{5 \sqrt{2-\sqrt{3}} b \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(7*a + 5*b*x))/(27*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (2*x*(a + b*x))/(9*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 +
x^3)) - (10*b*(1 + x^3))/(27*Sqrt[1 + x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) + (5*Sqrt[2 - Sqrt[3]]*b*Sqrt[1
+ x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sq
rt[3]])/(9*3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2]) + (2*Sqrt[2 + Sqrt[3]]*(7*a + 5*(1 - S
qrt[3])*b)*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3]
+ x)], -7 - 4*Sqrt[3]])/(27*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

Rule 809

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

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1878

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{a+b x}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1+x^3} \int \frac{a+b x}{\left (1+x^3\right )^{5/2}} \, dx}{\sqrt{1+x} \sqrt{1-x+x^2}}\\ &=\frac{2 x (a+b x)}{9 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )}-\frac{\left (2 \sqrt{1+x^3}\right ) \int \frac{-\frac{7 a}{2}-\frac{5 b x}{2}}{\left (1+x^3\right )^{3/2}} \, dx}{9 \sqrt{1+x} \sqrt{1-x+x^2}}\\ &=\frac{2 x (7 a+5 b x)}{27 \sqrt{1+x} \sqrt{1-x+x^2}}+\frac{2 x (a+b x)}{9 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )}+\frac{\left (4 \sqrt{1+x^3}\right ) \int \frac{\frac{7 a}{4}-\frac{5 b x}{4}}{\sqrt{1+x^3}} \, dx}{27 \sqrt{1+x} \sqrt{1-x+x^2}}\\ &=\frac{2 x (7 a+5 b x)}{27 \sqrt{1+x} \sqrt{1-x+x^2}}+\frac{2 x (a+b x)}{9 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )}-\frac{\left (5 b \sqrt{1+x^3}\right ) \int \frac{1-\sqrt{3}+x}{\sqrt{1+x^3}} \, dx}{27 \sqrt{1+x} \sqrt{1-x+x^2}}+\frac{\left (\left (7 a+5 \left (1-\sqrt{3}\right ) b\right ) \sqrt{1+x^3}\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx}{27 \sqrt{1+x} \sqrt{1-x+x^2}}\\ &=\frac{2 x (7 a+5 b x)}{27 \sqrt{1+x} \sqrt{1-x+x^2}}+\frac{2 x (a+b x)}{9 \sqrt{1+x} \sqrt{1-x+x^2} \left (1+x^3\right )}-\frac{10 b \left (1+x^3\right )}{27 \sqrt{1+x} \left (1+\sqrt{3}+x\right ) \sqrt{1-x+x^2}}+\frac{5 \sqrt{2-\sqrt{3}} b \sqrt{1+x} \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1-x+x^2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (7 a+5 \left (1-\sqrt{3}\right ) b\right ) \sqrt{1+x} \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1-x+x^2}}\\ \end{align*}

Mathematica [C]  time = 1.87298, size = 435, normalized size = 1.24 \[ \frac{2 x \left (a \left (7 x^3+10\right )+b x \left (5 x^3+8\right )\right )}{27 \left (x^2-x+1\right )^{3/2} (x+1)^{3/2}}+\frac{(x+1)^{3/2} \left (\frac{\sqrt{2} \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} \left (14 i \sqrt{3} a+5 \left (3-i \sqrt{3}\right ) b\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{\sqrt{3}+3 i}}}{\sqrt{x+1}}\right ),\frac{\sqrt{3}+3 i}{-\sqrt{3}+3 i}\right )}{\sqrt{x+1}}-\frac{60 \sqrt{-\frac{i}{\sqrt{3}+3 i}} b \left (x^2-x+1\right )}{(x+1)^2}+\frac{15 i \sqrt{2} \left (\sqrt{3}+i\right ) b \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{162 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(b*x*(8 + 5*x^3) + a*(10 + 7*x^3)))/(27*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)) + ((1 + x)^(3/2)*((-60*Sqrt[(-
I)/(3*I + Sqrt[3])]*b*(1 - x + x^2))/(1 + x)^2 + ((15*I)*Sqrt[2]*(I + Sqrt[3])*b*Sqrt[(3*I + Sqrt[3] - (6*I)/(
1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*
I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (Sqrt[2]*((14*I)*Sqrt[3]*a +
5*(3 - I*Sqrt[3])*b)*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 +
x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sq
rt[3])])/Sqrt[1 + x]))/(162*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])

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Maple [B]  time = 0.08, size = 1152, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(1+x)^(5/2)/(x^2-x+1)^(5/2),x)

[Out]

-1/27*(7*I*3^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*a*(-2*
(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+5
*I*3^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*b*(-2*(1+x)/(-
3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)-21*Ellipt
icF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*a*(-2*(1+x)/(-3+I*3^(1/2)))^(1/
2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+15*EllipticF((-2*(1+x)/(-3
+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*b*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*
x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)-30*EllipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2
),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^3*b*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3
))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+7*I*3^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+
I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*a*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I
*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+5*I*3^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(
I*3^(1/2)+3))^(1/2))*b*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x
-1)/(-3+I*3^(1/2)))^(1/2)-10*b*x^5-21*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*
((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)
+3))^(1/2))*a+15*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-
3+I*3^(1/2)))^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b-30*(-2*
(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*E
llipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*b-14*a*x^4-16*b*x^2-20*a*x)/(x
^2-x+1)^(3/2)/(1+x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(1+x)^(5/2)/(x^2-x+1)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}}{x^{9} + 3 \, x^{6} + 3 \, x^{3} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)*sqrt(x^2 - x + 1)*sqrt(x + 1)/(x^9 + 3*x^6 + 3*x^3 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(1+x)**(5/2)/(x**2-x+1)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)), x)