3.4.40 \(\int \frac {1}{(a+b x^2)^{11/4} (c+d x^2)^2} \, dx\) [340]

Optimal. Leaf size=419 \[ \frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (b c-a d)^4 x}+\frac {\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (b c-a d)^4 x} \]

[Out]

1/14*b*(7*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^2+a)^(7/4)+1/42*b*(-21*a^2*d^2-76*a*b*c*d+20*b^2*c^2)*x/a^2/c/(-a
*d+b*c)^3/(b*x^2+a)^(3/4)-1/2*d*x/c/(-a*d+b*c)/(b*x^2+a)^(7/4)/(d*x^2+c)+1/42*(-21*a^2*d^2-76*a*b*c*d+20*b^2*c
^2)*(1+b*x^2/a)^(3/4)*(cos(1/2*arctan(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x*b^(1/2)/a^(1/2)))*Elliptic
F(sin(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)/a^(3/2)/c/(-a*d+b*c)^3/(b*x^2+a)^(3/4)+1/4*a^(1/4)*d^2*(
-2*a*d+13*b*c)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*(-b*x^2/a)^(1/2)/c/(-a*d
+b*c)^4/x+1/4*a^(1/4)*d^2*(-2*a*d+13*b*c)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I
)*(-b*x^2/a)^(1/2)/c/(-a*d+b*c)^4/x

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Rubi [A]
time = 0.33, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {425, 541, 544, 239, 237, 410, 109, 418, 1232} \begin {gather*} \frac {b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{42 a^2 c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac {\sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{42 a^{3/2} c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac {\sqrt [4]{a} d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}+\frac {\sqrt [4]{a} d^2 \sqrt {-\frac {b x^2}{a}} (13 b c-2 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}-\frac {d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}+\frac {b x (7 a d+4 b c)}{14 a c \left (a+b x^2\right )^{7/4} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x]

[Out]

(b*(4*b*c + 7*a*d)*x)/(14*a*c*(b*c - a*d)^2*(a + b*x^2)^(7/4)) + (b*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*x)/
(42*a^2*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(7/4)*(c + d*x^2)) + (Sqrt[b]*
(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(42*
a^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) + (a^(1/4)*d^2*(13*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sq
rt[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)^4*x) + (a^(1/4)*d
^2*(13*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)^4*x)

Rule 109

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/4)), x_Symbol] :> Dist[-4, Subst[
Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e
, f}, x] && GtQ[-f/(d*e - c*f), 0]

Rule 237

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

Rule 239

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

Rule 410

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[Sqrt[(-b)*(x^2/a)]/(2*x), Subst[I
nt[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0]

Rule 418

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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

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 544

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 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], 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 )^{11/4} \left (c+d x^2\right )^2} \, dx &=-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-\frac {9}{2} b d x^2}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-10 b^2 c^2+28 a b c d-7 a^2 d^2\right )-\frac {5}{4} b d (4 b c+7 a d) x^2}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx}{7 a c (b c-a d)^2}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {2 \int \frac {\frac {1}{4} \left (10 b^3 c^3-38 a b^2 c^2 d+126 a^2 b c d^2-21 a^3 d^3\right )+\frac {1}{8} b d \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{21 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\left (d^2 (13 b c-2 a d)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^3}+\frac {\left (b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{84 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\left (d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {b x}{a}} (a+b x)^{3/4} (c+d x)} \, dx,x,x^2\right )}{8 c (b c-a d)^3 x}+\frac {\left (b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{84 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {\left (d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\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)^3 x}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac {\left (d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^4 x}+\frac {\left (d^2 (13 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^4 x}\\ &=\frac {b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac {b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} d^2 (13 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)^4 x}+\frac {\sqrt [4]{a} d^2 (13 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)^4 x}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.67, size = 550, normalized size = 1.31 \begin {gather*} \frac {\frac {b d \left (-20 b^2 c^2+76 a b c d+21 a^2 d^2\right ) x^3 \left (1+\frac {b x^2}{a}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{(-b c+a d)^3}+\frac {6 c \left (-6 a c x \left (42 a^4 d^3+63 a^3 b d^2 \left (-2 c+d x^2\right )-10 b^4 c^2 x^2 \left (3 c+2 d x^2\right )+a^2 b^2 d \left (126 c^2-38 c d x^2+21 d^2 x^4\right )+2 a b^3 c \left (-21 c^2+41 c d x^2+38 d^2 x^4\right )\right ) F_1\left (\frac {1}{2};\frac {3}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^3 \left (21 a^4 d^3+42 a^3 b d^3 x^2-20 b^4 c^2 x^2 \left (c+d x^2\right )+4 a b^3 c \left (-8 c^2+11 c d x^2+19 d^2 x^4\right )+a^2 b^2 d \left (88 c^2+88 c d x^2+21 d^2 x^4\right )\right ) \left (4 a d F_1\left (\frac {3}{2};\frac {3}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{(b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c F_1\left (\frac {1}{2};\frac {3}{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 {3}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c F_1\left (\frac {3}{2};\frac {7}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}}{252 a^2 c^2 \left (a+b x^2\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x]

[Out]

((b*d*(-20*b^2*c^2 + 76*a*b*c*d + 21*a^2*d^2)*x^3*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a
), -((d*x^2)/c)])/(-(b*c) + a*d)^3 + (6*c*(-6*a*c*x*(42*a^4*d^3 + 63*a^3*b*d^2*(-2*c + d*x^2) - 10*b^4*c^2*x^2
*(3*c + 2*d*x^2) + a^2*b^2*d*(126*c^2 - 38*c*d*x^2 + 21*d^2*x^4) + 2*a*b^3*c*(-21*c^2 + 41*c*d*x^2 + 38*d^2*x^
4))*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^3*(21*a^4*d^3 + 42*a^3*b*d^3*x^2 - 20*b^4*c^2*x
^2*(c + d*x^2) + 4*a*b^3*c*(-8*c^2 + 11*c*d*x^2 + 19*d^2*x^4) + a^2*b^2*d*(88*c^2 + 88*c*d*x^2 + 21*d^2*x^4))*
(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a),
-((d*x^2)/c)])))/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)*(6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*
x^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[3/2, 7/4, 1, 5/2
, -((b*x^2)/a), -((d*x^2)/c)]))))/(252*a^2*c^2*(a + b*x^2)^(3/4))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {11}{4}} \left (d \,x^{2}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^(11/4)/(d*x^2+c)^2,x)

[Out]

int(1/(b*x^2+a)^(11/4)/(d*x^2+c)^2,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(1/(b*x^2+a)^(11/4)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(11/4)*(d*x^2 + c)^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(11/4)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{2}\right )^{\frac {11}{4}} \left (c + d x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**(11/4)/(d*x**2+c)**2,x)

[Out]

Integral(1/((a + b*x**2)**(11/4)*(c + d*x**2)**2), x)

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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(1/(b*x^2+a)^(11/4)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(11/4)*(d*x^2 + c)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (b\,x^2+a\right )}^{11/4}\,{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)^(11/4)*(c + d*x^2)^2),x)

[Out]

int(1/((a + b*x^2)^(11/4)*(c + d*x^2)^2), x)

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