3.844 \(\int \frac{(e x)^m}{(a+b x^4) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=81 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e (m+1) \sqrt{c+d x^4}} \]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*AppellF1[(1 + m)/4, 1, 1/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*(1
+ m)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.0552466, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {511, 510} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};1,\frac{1}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*AppellF1[(1 + m)/4, 1, 1/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a*e*(1
+ m)*Sqrt[c + d*x^4])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 510

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

Rubi steps

\begin{align*} \int \frac{(e x)^m}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx &=\frac{\sqrt{1+\frac{d x^4}{c}} \int \frac{(e x)^m}{\left (a+b x^4\right ) \sqrt{1+\frac{d x^4}{c}}} \, dx}{\sqrt{c+d x^4}}\\ &=\frac{(e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} F_1\left (\frac{1+m}{4};1,\frac{1}{2};\frac{5+m}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e (1+m) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0950747, size = 125, normalized size = 1.54 \[ \frac{x \sqrt{c+d x^4} (e x)^m \left (b c F_1\left (\frac{m+1}{4};-\frac{1}{2},1;\frac{m+5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-a d \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )\right )}{a c (m+1) \sqrt{\frac{d x^4}{c}+1} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^m/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

(x*(e*x)^m*Sqrt[c + d*x^4]*(b*c*AppellF1[(1 + m)/4, -1/2, 1, (5 + m)/4, -((d*x^4)/c), -((b*x^4)/a)] - a*d*Hype
rgeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)]))/(a*c*(b*c - a*d)*(1 + m)*Sqrt[1 + (d*x^4)/c])

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{b{x}^{4}+a}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

int((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x)^m/((b*x^4 + a)*sqrt(d*x^4 + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{4} + c} \left (e x\right )^{m}}{b d x^{8} +{\left (b c + a d\right )} x^{4} + a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^4 + c)*(e*x)^m/(b*d*x^8 + (b*c + a*d)*x^4 + a*c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral((e*x)**m/((a + b*x**4)*sqrt(c + d*x**4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x)^m/((b*x^4 + a)*sqrt(d*x^4 + c)), x)