3.3.67 \(\int \frac {(a+b x^3+c x^6)^p}{x^4} \, dx\) [267]

Optimal. Leaf size=164 \[ -\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3} \]

[Out]

-1/3*4^p*(c*x^6+b*x^3+a)^p*AppellF1(1-2*p,-p,-p,2-2*p,1/2*(-b-(-4*a*c+b^2)^(1/2))/c/x^3,1/2*(-b+(-4*a*c+b^2)^(
1/2))/c/x^3)/(1-2*p)/x^3/(((b+2*c*x^3-(-4*a*c+b^2)^(1/2))/c/x^3)^p)/(((b+2*c*x^3+(-4*a*c+b^2)^(1/2))/c/x^3)^p)

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Rubi [A]
time = 0.09, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1371, 772, 138} \begin {gather*} -\frac {4^p \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3 + c*x^6)^p/x^4,x]

[Out]

-1/3*(4^p*(a + b*x^3 + c*x^6)^p*AppellF1[1 - 2*p, -p, -p, 2*(1 - p), -1/2*(b - Sqrt[b^2 - 4*a*c])/(c*x^3), -1/
2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3)])/((1 - 2*p)*x^3*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^
2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(-(1/(d + e*x))^(2*p))*((a + b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q +
2*c*x)/(2*c*(d + e*x))))^p)), Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b - q)/(2*c)))*x, x]^p*Simp[1 -
(d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, 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] &&  !IntegerQ[p] && ILtQ[m, 0]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^p}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^p}{x^2} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{3} \left (2^{2 p} \left (\frac {1}{x^3}\right )^{2 p} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p\right ) \text {Subst}\left (\int x^{2-2 (1+p)} \left (1+\frac {\left (b-\sqrt {b^2-4 a c}\right ) x}{2 c}\right )^p \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) x}{2 c}\right )^p \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac {b-\sqrt {b^2-4 a c}}{2 c x^3},-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 162, normalized size = 0.99 \begin {gather*} \frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2-2 p;-\frac {b+\sqrt {b^2-4 a c}}{2 c x^3},\frac {-b+\sqrt {b^2-4 a c}}{2 c x^3}\right )}{3 (-1+2 p) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3 + c*x^6)^p/x^4,x]

[Out]

(4^p*(a + b*x^3 + c*x^6)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt
[b^2 - 4*a*c])/(2*c*x^3)])/(3*(-1 + 2*p)*x^3*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*
a*c] + 2*c*x^3)/(c*x^3))^p)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^6+b*x^3+a)^p/x^4,x)

[Out]

int((c*x^6+b*x^3+a)^p/x^4,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**6+b*x**3+a)**p/x**4,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^6+b*x^3+a)^p/x^4,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^p}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3 + c*x^6)^p/x^4,x)

[Out]

int((a + b*x^3 + c*x^6)^p/x^4, x)

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