\(\int \frac {(a+b x^2)^{7/4}}{c+d x^2} \, dx\) [321]

Optimal result

Integrand size = 21, antiderivative size = 362 \[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\frac {6 a b x}{5 d \sqrt [4]{a+b x^2}}-\frac {2 b (b c-a d) x}{d^2 \sqrt [4]{a+b x^2}}+\frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d}-\frac {6 a^{3/2} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac {2 \sqrt {a} \sqrt {b} (b c-a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (-b c+a d)^{3/2} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x}-\frac {\sqrt [4]{a} (-b c+a d)^{3/2} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x} \]

[Out]

6/5*a*b*x/d/(b*x^2+a)^(1/4)-2*b*(-a*d+b*c)*x/d^2/(b*x^2+a)^(1/4)+2/5*b*x*(b*x^2+a)^(3/4)/d-6/5*a^(3/2)*(1+b*x^
2/a)^(1/4)*(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)))*EllipticE(sin(1/2*a
rctan(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)/d/(b*x^2+a)^(1/4)+2*(-a*d+b*c)*(1+b*x^2/a)^(1/4)*(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)))*EllipticE(sin(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(
1/2))*a^(1/2)*b^(1/2)/d^2/(b*x^2+a)^(1/4)+a^(1/4)*(a*d-b*c)^(3/2)*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)/d^(5/2)/x-a^(1/4)*(a*d-b*c)^(3/2)*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)/d^(5/2)/x

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {411, 201, 235, 233, 202, 408, 504, 1232} \[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=-\frac {6 a^{3/2} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (a d-b c)^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (a d-b c)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^{5/2} x}+\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} (b c-a d) E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}-\frac {2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac {6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]

[In]

Int[(a + b*x^2)^(7/4)/(c + d*x^2),x]

[Out]

(6*a*b*x)/(5*d*(a + b*x^2)^(1/4)) - (2*b*(b*c - a*d)*x)/(d^2*(a + b*x^2)^(1/4)) + (2*b*x*(a + b*x^2)^(3/4))/(5
*d) - (6*a^(3/2)*Sqrt[b]*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*d*(a + b*x^2)^(
1/4)) + (2*Sqrt[a]*Sqrt[b]*(b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(d^2
*(a + b*x^2)^(1/4)) + (a^(1/4)*(-(b*c) + a*d)^(3/2)*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])/(d^(5/2)*x) - (a^(1/4)*(-(b*c) + a*d)^(3/2)*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])/(d^(5/2)*x)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 202

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

Rule 233

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 235

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 408

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 411

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

Rule 504

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), Int[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 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*} \text {integral}& = \frac {b \int \left (a+b x^2\right )^{3/4} \, dx}{d}-\frac {(b c-a d) \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx}{d} \\ & = \frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d}+\frac {(3 a b) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{5 d}-\frac {(b (b c-a d)) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{d^2} \\ & = \frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d}+\frac {\left (2 (b c-a d)^2 \sqrt {-\frac {b x^2}{a}}\right ) \text {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 )}{d^2 x}+\frac {\left (3 a b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{5 d \sqrt [4]{a+b x^2}}-\frac {\left (b (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}{d^2 \sqrt [4]{a+b x^2}} \\ & = \frac {6 a b x}{5 d \sqrt [4]{a+b x^2}}-\frac {2 b (b c-a d) x}{d^2 \sqrt [4]{a+b x^2}}+\frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d}-\frac {\left ((b c-a d)^2 \sqrt {-\frac {b x^2}{a}}\right ) \text {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 )}{d^{5/2} x}+\frac {\left ((b c-a d)^2 \sqrt {-\frac {b x^2}{a}}\right ) \text {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 )}{d^{5/2} x}-\frac {\left (3 a b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{5 d \sqrt [4]{a+b x^2}}+\frac {\left (b (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}{d^2 \sqrt [4]{a+b x^2}} \\ & = \frac {6 a b x}{5 d \sqrt [4]{a+b x^2}}-\frac {2 b (b c-a d) x}{d^2 \sqrt [4]{a+b x^2}}+\frac {2 b x \left (a+b x^2\right )^{3/4}}{5 d}-\frac {6 a^{3/2} \sqrt {b} \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 )}{5 d \sqrt [4]{a+b x^2}}+\frac {2 \sqrt {a} \sqrt {b} (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 )}{d^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (-b c+a d)^{3/2} \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 )}{d^{5/2} x}-\frac {\sqrt [4]{a} (-b c+a d)^{3/2} \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 )}{d^{5/2} x} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 9.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\frac {x \left (\frac {b (-5 b c+8 a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (-3 a c \left (5 a^2 d+2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{15 d \sqrt [4]{a+b x^2}} \]

[In]

Integrate[(a + b*x^2)^(7/4)/(c + d*x^2),x]

[Out]

(x*((b*(-5*b*c + 8*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 +
(6*(-3*a*c*(5*a^2*d + 2*a*b*d*x^2 + 2*b^2*x^2*(c + d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/
c)] + b*x^2*(a + b*x^2)*(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*Appell
F1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6*a*c*AppellF1[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)])))))/(15*d*(a + b*x^2)^(1/4))

Maple [F]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {7}{4}}}{d \,x^{2}+c}d x\]

[In]

int((b*x^2+a)^(7/4)/(d*x^2+c),x)

[Out]

int((b*x^2+a)^(7/4)/(d*x^2+c),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\text {Timed out} \]

[In]

integrate((b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {7}{4}}}{c + d x^{2}}\, dx \]

[In]

integrate((b*x**2+a)**(7/4)/(d*x**2+c),x)

[Out]

Integral((a + b*x**2)**(7/4)/(c + d*x**2), x)

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{4}}}{d x^{2} + c} \,d x } \]

[In]

integrate((b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(7/4)/(d*x^2 + c), x)

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{4}}}{d x^{2} + c} \,d x } \]

[In]

integrate((b*x^2+a)^(7/4)/(d*x^2+c),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(7/4)/(d*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/4}}{d\,x^2+c} \,d x \]

[In]

int((a + b*x^2)^(7/4)/(c + d*x^2),x)

[Out]

int((a + b*x^2)^(7/4)/(c + d*x^2), x)