3.4.20 \(\int (a+b x^n)^p (c+d x^n)^{-1-\frac {1}{n}-p} \, dx\) [320]

Optimal. Leaf size=93 \[ \frac {x \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{c} \]

[Out]

x*(a+b*x^n)^p*(c+d*x^n)^(-1/n-p)*hypergeom([-p, 1/n],[1+1/n],-(-a*d+b*c)*x^n/a/(c+d*x^n))/c/((c*(a+b*x^n)/a/(c
+d*x^n))^p)

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Rubi [A]
time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {388} \begin {gather*} \frac {x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac {1}{n}-p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^(-1 - n^(-1) - p),x]

[Out]

(x*(a + b*x^n)^p*(c + d*x^n)^(-n^(-1) - p)*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c
+ d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p)

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a
+ b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n)^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(
a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rubi steps

\begin {align*} \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, dx &=\frac {x \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{c}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 94, normalized size = 1.01 \begin {gather*} \frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^{-\frac {1+n p}{n}} \left (1+\frac {d x^n}{c}\right )^p \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{c} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^n)^p*(c + d*x^n)^(-1 - n^(-1) - p),x]

[Out]

(x*(a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), ((-(b*c) + a*d)*x^n)/(a*(c + d*x^
n))])/(c*(1 + (b*x^n)/a)^p*(c + d*x^n)^((1 + n*p)/n))

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{-1-\frac {1}{n}-p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),x)

[Out]

int((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),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((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 1), 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((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p/(d*x^n + c)^((n*p + n + 1)/n), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**p*(c+d*x**n)**(-1-1/n-p),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5010 deep

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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((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p/(c + d*x^n)^(p + 1/n + 1),x)

[Out]

int((a + b*x^n)^p/(c + d*x^n)^(p + 1/n + 1), x)

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