3.852 \(\int \frac{(e x)^m}{(a+b x^4)^3 (c+d x^4)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 c e (m+1) \sqrt{c+d x^4}} \]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*AppellF1[(1 + m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^3*c*
e*(1 + m)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.0618125, antiderivative size = 84, 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} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 c e (m+1) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m/((a + b*x^4)^3*(c + d*x^4)^(3/2)),x]

[Out]

((e*x)^(1 + m)*Sqrt[1 + (d*x^4)/c]*AppellF1[(1 + m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^3*c*
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 )^3 \left (c+d x^4\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{d x^4}{c}} \int \frac{(e x)^m}{\left (a+b x^4\right )^3 \left (1+\frac{d x^4}{c}\right )^{3/2}} \, dx}{c \sqrt{c+d x^4}}\\ &=\frac{(e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} F_1\left (\frac{1+m}{4};3,\frac{3}{2};\frac{5+m}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 c e (1+m) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [A]  time = 0.078791, size = 77, normalized size = 0.92 \[ \frac{x \left (\frac{d x^4}{c}+1\right )^{3/2} (e x)^m F_1\left (\frac{m+1}{4};3,\frac{3}{2};\frac{m+5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^3 (m+1) \left (c+d x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m/((a + b*x^4)^3*(c + d*x^4)^(3/2)),x]

[Out]

(x*(e*x)^m*(1 + (d*x^4)/c)^(3/2)*AppellF1[(1 + m)/4, 3, 3/2, (5 + m)/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^3*(1 +
m)*(c + d*x^4)^(3/2))

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ( d{x}^{4}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x)

[Out]

int((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),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 )}^{3}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)), 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^{3} d^{2} x^{20} +{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{16} +{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{12} +{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{8} + a^{3} c^{2} +{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^4 + c)*(e*x)^m/(b^3*d^2*x^20 + (2*b^3*c*d + 3*a*b^2*d^2)*x^16 + (b^3*c^2 + 6*a*b^2*c*d + 3*a
^2*b*d^2)*x^12 + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^8 + a^3*c^2 + (3*a^2*b*c^2 + 2*a^3*c*d)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m/(b*x**4+a)**3/(d*x**4+c)**(3/2),x)

[Out]

Timed out

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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 )}^{3}{\left (d x^{4} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m/(b*x^4+a)^3/(d*x^4+c)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x)^m/((b*x^4 + a)^3*(d*x^4 + c)^(3/2)), x)