3.4.23 \(\int \frac {(a+b x^2)^{3/4}}{c+d x^2} \, dx\) [323]

3.4.23.1 Optimal result

Integrand size = 21, antiderivative size = 244 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\frac {2 b x}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-b c+a d} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x}-\frac {\sqrt [4]{a} \sqrt {-b c+a d} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x} \]

output

2*b*x/d/(b*x^2+a)^(1/4)-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*arctan 
(x*b^(1/2)/a^(1/2))),2^(1/2))*a^(1/2)*b^(1/2)/d/(b*x^2+a)^(1/4)+a^(1/4)*El 
lipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*(a*d- 
b*c)^(1/2)*(-b*x^2/a)^(1/2)/d^(3/2)/x-a^(1/4)*EllipticPi((b*x^2+a)^(1/4)/a 
^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)*(a*d-b*c)^(1/2)*(-b*x^2/a)^(1/2) 
/d^(3/2)/x
 

3.4.23.2 Mathematica [C] (warning: unable to verify)

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

Time = 9.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\frac {6 a c x \left (a+b x^2\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},-\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )} \]

input

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

output

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

3.4.23.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {301, 227, 225, 212, 310, 993, 1542}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {b \int \frac {1}{\sqrt [4]{b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{d \sqrt [4]{a+b x^2}}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{d \sqrt [4]{a+b x^2}}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{d x}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{d x}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{d x}\)

input

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

output

(b*(1 + (b*x^2)/a)^(1/4)*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*Ellipti 
cE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/Sqrt[b]))/(d*(a + b*x^2)^(1/4)) - (2 
*(b*c - a*d)*Sqrt[-((b*x^2)/a)]*((a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[d])/S 
qrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqr 
t[-(b*c) + a*d]) - (a^(1/4)*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d 
], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(2*Sqrt[d]*Sqrt[-(b*c) + a*d])) 
)/(d*x)
 

3.4.23.3.1 Defintions of rubi rules used

rule 212

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 225

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

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(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] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[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 993

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]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[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 1542

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]
 

3.4.23.4 Maple [F]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{d \,x^{2}+c}d x\]

input

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

output

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

3.4.23.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.4.23.6 Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{4}}}{c + d x^{2}}\, dx \]

input

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

output

Integral((a + b*x**2)**(3/4)/(c + d*x**2), x)
 

3.4.23.7 Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{d x^{2} + c} \,d x } \]

input

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

output

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

3.4.23.8 Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{d x^{2} + c} \,d x } \]

input

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

output

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

3.4.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/4}}{d\,x^2+c} \,d x \]

input

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

output

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