### 3.2492 $$\int \frac{1}{(a+b x+c x^2)^{7/3}} \, dx$$

Optimal. Leaf size=993 $\text{result too large to display}$

[Out]

(-3*(b + 2*c*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) + (15*c*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*(a + b*x +
c*x^2)^(1/3)) - (15*c^(4/3)*(b + 2*c*x))/(2^(1/3)*(b^2 - 4*a*c)^2*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)
*c^(1/3)*(a + b*x + c*x^2)^(1/3))) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1
/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)
^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a +
b*x + c*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)
^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^
(1/3)*(b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*
x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]) - (5*2^(1
/6)*3^(3/4)*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3)
- 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1
+ Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*
(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^
(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/((b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*(
(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c
^(1/3)*(a + b*x + c*x^2)^(1/3))^2])

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Rubi [A]  time = 0.953901, antiderivative size = 993, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.357, Rules used = {623, 325, 303, 218, 1877} $\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right ) c^{4/3}}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}-\frac{5 \sqrt [6]{2} 3^{3/4} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right ) c^{4/3}}{\left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}-\frac{15 (b+2 c x) c^{4/3}}{\sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}+\frac{15 (b+2 c x) c}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(-7/3),x]

[Out]

(-3*(b + 2*c*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) + (15*c*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*(a + b*x +
c*x^2)^(1/3)) - (15*c^(4/3)*(b + 2*c*x))/(2^(1/3)*(b^2 - 4*a*c)^2*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)
*c^(1/3)*(a + b*x + c*x^2)^(1/3))) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1
/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)
^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a +
b*x + c*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)
^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*2^
(1/3)*(b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*
x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]) - (5*2^(1
/6)*3^(3/4)*c^(4/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3)
- 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1
+ Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*
(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^
(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/((b^2 - 4*a*c)^(5/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*(
(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c
^(1/3)*(a + b*x + c*x^2)^(1/3))^2])

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

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

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{1}{\left (a+b x+c x^2\right )^{7/3}} \, dx &=\frac{\left (3 \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{b+2 c x}\\ &=-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}-\frac{\left (15 c \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 \left (b^2-4 a c\right ) (b+2 c x)}\\ &=-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac{15 c (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac{\left (15 c^2 \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^2 (b+2 c x)}\\ &=-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac{15 c (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac{\left (15 c^{5/3} \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} x}{\sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2^{2/3} \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac{\left (15 c^{5/3} \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{\sqrt [6]{2} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^{5/3} (b+2 c x)}\\ &=-\frac{3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac{15 c (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac{15 c^{4/3} (b+2 c x)}{\sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}+\frac{15 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{4/3} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{5 \sqrt [6]{2} 3^{3/4} c^{4/3} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt{3}\right )}{\left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.20328, size = 138, normalized size = 0.14 $\frac{(b+2 c x) \left (3 \sqrt [3]{2} \left (2 c \left (7 a+5 c x^2\right )-b^2+10 b c x\right )-20 c (a+x (b+c x)) \sqrt [3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(-7/3),x]

[Out]

((b + 2*c*x)*(3*2^(1/3)*(-b^2 + 10*b*c*x + 2*c*(7*a + 5*c*x^2)) - 20*c*(a + x*(b + c*x))*((c*(a + x*(b + c*x))
)/(-b^2 + 4*a*c))^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(4*2^(1/3)*(b^2 - 4*a*
c)^2*(a + x*(b + c*x))^(4/3))

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Maple [F]  time = 2.279, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+a)^(7/3),x)

[Out]

int(1/(c*x^2+b*x+a)^(7/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(-7/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{2}{3}}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(2/3)/(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*
x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{7}{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x+a)**(7/3),x)

[Out]

Integral((a + b*x + c*x**2)**(-7/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(-7/3), x)