3.4.31 \(\int \frac {(a+b x^2)^{7/4}}{(c+d x^2)^2} \, dx\) [331]

3.4.31.1 Optimal result

Integrand size = 21, antiderivative size = 340 \[ \int \frac {\left (a+b x^2\right )^{7/4}}{\left (c+d x^2\right )^2} \, dx=\frac {b (5 b c-a d) x}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} (5 b c-a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-b c+a d} (5 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{5/2} x}-\frac {\sqrt [4]{a} \sqrt {-b c+a d} (5 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{5/2} x} \]

output

1/2*b*(-a*d+5*b*c)*x/c/d^2/(b*x^2+a)^(1/4)-1/2*(-a*d+b*c)*x*(b*x^2+a)^(3/4 
)/c/d/(d*x^2+c)-1/2*(-a*d+5*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)/c/d^2/(b*x^2+a)^(1/4) 
+1/4*a^(1/4)*(2*a*d+5*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)*(a*d-b*c)^(1/2)*(-b*x^2/a)^(1/2)/c/d^(5/2)/x-1/4*a 
^(1/4)*(2*a*d+5*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)*(a*d-b*c)^(1/2)*(-b*x^2/a)^(1/2)/c/d^(5/2)/x
 

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

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

Time = 10.35 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^{7/4}}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (-b (-5 b c+a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (-6 a c \left (2 a^2 d-b^2 c x^2+a b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+(-b c+a d) x^2 \left (a+b x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 c^2 d \sqrt [4]{a+b x^2}} \]

input

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

output

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

3.4.31.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 317, 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 = {315, 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.

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

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

\(\Big \downarrow \) 227

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

\(\Big \downarrow \) 225

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

\(\Big \downarrow \) 212

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

\(\Big \downarrow \) 310

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {b \sqrt [4]{\frac {b x^2}{a}+1} (5 b c-a d) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) (2 a d+5 b c) \left (\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt {d} \sqrt {a d-b c}}\right )}{d x}}{4 c d}-\frac {x \left (a+b x^2\right )^{3/4} (b c-a d)}{2 c d \left (c+d x^2\right )}\)

input

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

output

-1/2*((b*c - a*d)*x*(a + b*x^2)^(3/4))/(c*d*(c + d*x^2)) + ((b*(5*b*c - a* 
d)*(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)*(5*b*c + 2*a*d)*Sqrt[-((b*x^2)/a)]*((a^(1/4)*EllipticPi[-((Sq 
rt[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])/S 
qrt[-(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*d)
 

3.4.31.3.1 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 315

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

3.4.31.4 Maple [F]

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

input

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

output

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

3.4.31.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.4.31.6 Sympy [F]

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

input

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

output

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

3.4.31.7 Maxima [F]

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

input

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

output

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

3.4.31.8 Giac [F]

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

input

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

output

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

3.4.31.9 Mupad [F(-1)]

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

input

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

output

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