3.212 \(\int \frac{(a+b x^4)^{5/4}}{(c+d x^4)^2} \, dx\)

Optimal. Leaf size=298 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}-\frac{x \sqrt [4]{a+b x^4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d*(c + d*x^4)) + (Sqrt[a]*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[
ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*c*d*(a + b*x^4)^(3/4)) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a +
b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4
)*c^2*d) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]),
ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*d)

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Rubi [A]  time = 0.239795, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {413, 529, 237, 335, 275, 231, 407, 409, 1218} \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (3 a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}-\frac{x \sqrt [4]{a+b x^4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/(c + d*x^4)^2,x]

[Out]

-((b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d*(c + d*x^4)) + (Sqrt[a]*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[
ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*c*d*(a + b*x^4)^(3/4)) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a +
b*x^4]*EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4
)*c^2*d) + ((2*b*c + 3*a*d)*Sqrt[a/(a + b*x^4)]*Sqrt[a + b*x^4]*EllipticPi[Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c]),
ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1])/(8*b^(1/4)*c^2*d)

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 529

Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[(b*e - a*f)/(
b*c - a*d), Int[1/(a + b*x^4)^(3/4), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4),
x], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 407

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)],
Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac{\int \frac{a (b c+3 a d)+2 b (b c+a d) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}-\frac{(a b) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c d}-\frac{(-2 b c-3 a d) \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}-\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{4 c d \left (a+b x^4\right )^{3/4}}-\frac{\left ((-2 b c-3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{4 c d}\\ &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{4 c d \left (a+b x^4\right )^{3/4}}-\frac{\left ((-2 b c-3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 d}-\frac{\left ((-2 b c-3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 d}\\ &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac{(2 b c+3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{(2 b c+3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{8 c d \left (a+b x^4\right )^{3/4}}\\ &=-\frac{(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac{\sqrt{a} b^{3/2} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac{(2 b c+3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac{(2 b c+3 a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}\\ \end{align*}

Mathematica [C]  time = 0.324814, size = 341, normalized size = 1.14 \[ \frac{x \left (\frac{5 c \left (x^4 \left (a+b x^4\right ) (a d-b c) \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c \left (4 a^2 d+a b d x^4-b^2 c x^4\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+2 b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} (a d+b c) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{20 c^2 d \left (a+b x^4\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^4)^(5/4)/(c + d*x^4)^2,x]

[Out]

(x*(2*b*(b*c + a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + (5*c*(-
5*a*c*(4*a^2*d - b^2*c*x^4 + a*b*d*x^4)*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + (-(b*c) + a*d
)*x^4*(a + b*x^4)*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1,
9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)
] + x^4*(4*a*d*AppellF1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -((b*
x^4)/a), -((d*x^4)/c)])))))/(20*c^2*d*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/(d*x^4+c)^2,x)

[Out]

int((b*x^4+a)^(5/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/(d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)/(d*x^4 + c)^2, x)

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Fricas [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((b*x^4+a)^(5/4)/(d*x^4+c)^2,x, algorithm="fricas")

[Out]

Timed out

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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((b*x**4+a)**(5/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/(d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/(d*x^4 + c)^2, x)