\(\int \frac {(a+b x^2)^{3/4}}{(c+d x^2)^2} \, dx\) [414]

Optimal result

Integrand size = 21, antiderivative size = 309 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x}-\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x} \] Output:

-1/2*b*x/c/d/(b*x^2+a)^(1/4)+1/2*x*(b*x^2+a)^(3/4)/c/(d*x^2+c)+1/2*a^(1/2) 
*b^(1/2)*(1+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^ 
(1/2))/c/d/(b*x^2+a)^(1/4)+1/4*a^(1/4)*(2*a*d+b*c)*(-b*x^2/a)^(1/2)*Ellipt 
icPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c/d^(3/2) 
/(a*d-b*c)^(1/2)/x-1/4*a^(1/4)*(2*a*d+b*c)*(-b*x^2/a)^(1/2)*EllipticPi((b* 
x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/c/d^(3/2)/(a*d-b*c 
)^(1/2)/x
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.25 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (-\frac {b x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2}+\frac {6 \left (\frac {a+b x^2}{c}-\frac {6 a^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}\right )}{c+d x^2}\right )}{12 \sqrt [4]{a+b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {314, 27, 405, 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.

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

\(\displaystyle \frac {\frac {(2 a d+b c) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {b \int \frac {1}{\sqrt [4]{b x^2+a}}dx}{d}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\frac {(2 a d+b c) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {\frac {(2 a d+b c) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {\frac {(2 a d+b c) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx}{d}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {\frac {2 \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{d x}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {\frac {2 \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{d x}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {2 \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{d x}-\frac {b \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}}{4 c}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

rule 405

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

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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