3.15.45 \(\int \frac {x^m}{a-b x^7} \, dx\) [1445]

Optimal. Leaf size=38 \[ \frac {x^{1+m} \, _2F_1\left (1,\frac {1+m}{7};\frac {8+m}{7};\frac {b x^7}{a}\right )}{a (1+m)} \]

[Out]

x^(1+m)*hypergeom([1, 1/7+1/7*m],[8/7+1/7*m],b*x^7/a)/a/(1+m)

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {371} \begin {gather*} \frac {x^{m+1} \, _2F_1\left (1,\frac {m+1}{7};\frac {m+8}{7};\frac {b x^7}{a}\right )}{a (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m/(a - b*x^7),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, (1 + m)/7, (8 + m)/7, (b*x^7)/a])/(a*(1 + m))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {x^m}{a-b x^7} \, dx &=\frac {x^{1+m} \, _2F_1\left (1,\frac {1+m}{7};\frac {8+m}{7};\frac {b x^7}{a}\right )}{a (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 40, normalized size = 1.05 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (1,\frac {1+m}{7};1+\frac {1+m}{7};\frac {b x^7}{a}\right )}{a (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a - b*x^7),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, (1 + m)/7, 1 + (1 + m)/7, (b*x^7)/a])/(a*(1 + m))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{-b \,x^{7}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(-b*x^7+a),x)

[Out]

int(x^m/(-b*x^7+a),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(x^m/(-b*x^7+a),x, algorithm="maxima")

[Out]

-integrate(x^m/(b*x^7 - a), x)

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Fricas [F]
time = 0.36, size = 18, normalized size = 0.47 \begin {gather*} {\rm integral}\left (-\frac {x^{m}}{b x^{7} - a}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(-b*x^7+a),x, algorithm="fricas")

[Out]

integral(-x^m/(b*x^7 - a), 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(x**m/(-b*x**7+a),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(x^m/(-b*x^7+a),x, algorithm="giac")

[Out]

integrate(-x^m/(b*x^7 - a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m}{a-b\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a - b*x^7),x)

[Out]

int(x^m/(a - b*x^7), x)

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