3.337 \(\int \frac{1}{(a+b x^2)^{5/4} (c+d x^2)^2} \, dx\)

Optimal. Leaf size=314 \[ -\frac{d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (a d+4 b c) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} c \sqrt [4]{a+b x^2} (b c-a d)^2}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}} \]

[Out]

-(d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + a*d)*(1 + (b*x^2)/a)^(1/4)*Elliptic
E[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*Sqrt[a]*c*(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*Sqrt[d]*(7*b*c -
2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4
)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x) + (a^(1/4)*Sqrt[d]*(7*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a
]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x)

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Rubi [A]  time = 0.395199, antiderivative size = 314, 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 = {414, 527, 530, 229, 227, 196, 399, 490, 1218} \[ -\frac{d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (a d+4 b c) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} c \sqrt [4]{a+b x^2} (b c-a d)^2}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]

[Out]

-(d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + a*d)*(1 + (b*x^2)/a)^(1/4)*Elliptic
E[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*Sqrt[a]*c*(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*Sqrt[d]*(7*b*c -
2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4
)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x) + (a^(1/4)*Sqrt[d]*(7*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a
]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*(-(b*c) + a*d)^(5/2)*x)

Rule 414

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

Rule 527

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

Rule 530

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

Rule 229

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Dist[(1 + (b*x^2)/a)^(1/4)/(a + b*x^2)^(1/4), Int[1/(1 + (b*x^2
)/a)^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*x)/(a + b*x^2)^(1/4), x] - Dist[a, Int[1/(a + b*x^2)^(5
/4), x], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 196

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(5/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 399

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

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), 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{1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx &=-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d-\frac{3}{2} b d x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac{b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac{\int \frac{\frac{1}{2} \left (2 b^2 c^2+4 a b c d-a^2 d^2\right )+\frac{1}{4} b d (4 b c+a d) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{a c (b c-a d)^2}\\ &=\frac{b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac{(d (7 b c-2 a d)) \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^2}-\frac{(b (4 b c+a d)) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{4 a c (b c-a d)^2}\\ &=\frac{b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac{\left (d (7 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c (b c-a d)^2 x}-\frac{\left (b (4 b c+a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac{\left (\sqrt{d} (7 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}-\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^2 x}-\frac{\left (\sqrt{d} (7 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}+\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^2 x}+\frac{\left (b (4 b c+a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac{\sqrt{b} (4 b c+a d) \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac{\sqrt [4]{a} \sqrt{d} (7 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{5/2} x}+\frac{\sqrt [4]{a} \sqrt{d} (7 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{5/2} x}\\ \end{align*}

Mathematica [C]  time = 0.498465, size = 380, normalized size = 1.21 \[ \frac{x \left (\frac{c \left (36 a c \left (2 a^2 d^2+a b d \left (d x^2-4 c\right )+2 b^2 c \left (c+2 d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-6 x^2 \left (a^2 d^2+a b d^2 x^2+4 b^2 c \left (c+d x^2\right )\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}-b d x^2 \sqrt [4]{\frac{b x^2}{a}+1} (a d+4 b c) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{12 a c^2 \sqrt [4]{a+b x^2} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]

[Out]

(x*(-(b*d*(4*b*c + a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]) + (c
*(36*a*c*(2*a^2*d^2 + a*b*d*(-4*c + d*x^2) + 2*b^2*c*(c + 2*d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -
((d*x^2)/c)] - 6*x^2*(a^2*d^2 + a*b*d^2*x^2 + 4*b^2*c*(c + d*x^2))*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)
/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(6*a*c*AppellF
1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2
)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(12*a*c^2*(b*c - a*d)^2*(a + b*x^2)^(1
/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x)

[Out]

int(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + 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(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**(5/4)/(d*x**2+c)**2,x)

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^2), x)