3.2.51 \(\int \frac {(d+e x^n)^q}{x^3 (a+b x^n+c x^{2 n})} \, dx\) [151]

Optimal. Leaf size=210 \[ \frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x^2}+\frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x^2} \]

[Out]

c*(d+e*x^n)^q*AppellF1(-2/n,1,-q,(-2+n)/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)),-e*x^n/d)/x^2/((1+e*x^n/d)^q)/(b^2-4
*a*c-b*(-4*a*c+b^2)^(1/2))+c*(d+e*x^n)^q*AppellF1(-2/n,1,-q,(-2+n)/n,-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)),-e*x^n/d)
/x^2/((1+e*x^n/d)^q)/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))

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Rubi [A]
time = 0.16, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1570, 525, 524} \begin {gather*} \frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}+\frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x^n)^q*AppellF1[-2/n, 1, -q, -((2 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/((b^2
- 4*a*c - b*Sqrt[b^2 - 4*a*c])*x^2*(1 + (e*x^n)/d)^q) + (c*(d + e*x^n)^q*AppellF1[-2/n, 1, -q, -((2 - n)/n), (
-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*x^2*(1 + (e*x^n)/d)^q)

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1570

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> With[{r = Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/r), Int[(f*x)^m*((d + e*x^n)^q/(b - r + 2*c*x^n)), x], x] - Dist[
2*(c/r), Int[(f*x)^m*((d + e*x^n)^q/(b + r + 2*c*x^n)), x], x]] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx &=\frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{x^3 \left (b-\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{x^3 \left (b+\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^n}{d}\right )^q}{x^3 \left (b-\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^n}{d}\right )^q}{x^3 \left (b+\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x^2}+\frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x^2}\\ \end {align*}

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Mathematica [F]
time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (d +e \,x^{n}\right )^{q}}{x^{3} \left (a +b \,x^{n}+c \,x^{2 n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d+e*x^n)^q/x^3/(a+b*x^n+c*x^(2*n)),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((d+e*x^n)^q/x^3/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

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

[Out]

integral((x^n*e + d)^q/(c*x^3*x^(2*n) + b*x^3*x^n + a*x^3), 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((d+e*x**n)**q/x**3/(a+b*x**n+c*x**(2*n)),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((d+e*x^n)^q/x^3/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate((x^n*e + d)^q/((c*x^(2*n) + b*x^n + a)*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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