3.4.22 \(\int \frac {(a+b x^2)^{5/4}}{c+d x^2} \, dx\) [322]

3.4.22.1 Optimal result

Integrand size = 21, antiderivative size = 302 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {2 a^{3/2} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {a} \sqrt {b} (b c-a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (b c-a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x}+\frac {\sqrt [4]{a} (b c-a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x} \]

output

2/3*b*x*(b*x^2+a)^(1/4)/d+2/3*a^(3/2)*(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)))*EllipticF(si 
n(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)/d/(b*x^2+a)^(3/4)-2*(-a* 
d+b*c)*(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)))*EllipticF(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)^(3/4)+a^(1/4)*(-a*d+b*c)*Elliptic 
Pi((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^2/x+a^(1/4)*(-a*d+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)/d^2/x
 

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

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

Time = 10.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\frac {x \left (\frac {b (-3 b c+4 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (-3 a c \left (3 a^2 d+2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{9 d \left (a+b x^2\right )^{3/4}} \]

input

Integrate[(a + b*x^2)^(5/4)/(c + d*x^2),x]
 

output

(x*((b*(-3*b*c + 4*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/ 
2, -((b*x^2)/a), -((d*x^2)/c)])/c + (6*(-3*a*c*(3*a^2*d + 2*a*b*d*x^2 + 2* 
b^2*x^2*(c + d*x^2))*AppellF1[1/2, 3/4, 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, 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)])))/((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)])) 
)))/(9*d*(a + b*x^2)^(3/4))
 

3.4.22.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {301, 211, 231, 229, 301, 231, 229, 312, 118, 25, 925, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx\)

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 231

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 231

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 312

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

\(\Big \downarrow \) 118

\(\displaystyle \frac {b \left (\frac {2 a^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {2}{3} x \sqrt [4]{a+b x^2}\right )}{d}-\frac {(b c-a d) \left (\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \int -\frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{d x}+\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d \left (a+b x^2\right )^{3/4}}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {2 a^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {2}{3} x \sqrt [4]{a+b x^2}\right )}{d}-\frac {(b c-a d) \left (\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \int \frac {1}{\sqrt {1-\frac {x^8}{a}} \left (d x^8+b c-a d\right )}d\sqrt [4]{b x^2+a}}{d x}\right )}{d}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {b \left (\frac {2 a^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {2}{3} x \sqrt [4]{a+b x^2}\right )}{d}-\frac {(b c-a d) \left (\frac {2 \sqrt {-\frac {b x^2}{a}} (b c-a d) \left (-\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}-\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^4}{\sqrt {a d-b c}}+1\right ) \sqrt {1-\frac {x^8}{a}}}d\sqrt [4]{b x^2+a}}{2 (b c-a d)}\right )}{d x}+\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d \left (a+b x^2\right )^{3/4}}\right )}{d}\)

\(\Big \downarrow \) 1542

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

input

Int[(a + b*x^2)^(5/4)/(c + d*x^2),x]
 

output

(b*((2*x*(a + b*x^2)^(1/4))/3 + (2*a^(3/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF 
[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a + b*x^2)^(3/4))))/d - (( 
b*c - a*d)*((2*Sqrt[a]*Sqrt[b]*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqr 
t[b]*x)/Sqrt[a]]/2, 2])/(d*(a + b*x^2)^(3/4)) + (2*(b*c - a*d)*Sqrt[-((b*x 
^2)/a)]*(-1/2*(a^(1/4)*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), 
 ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(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*(b*c - a*d))))/(d*x)))/d
 

3.4.22.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 118

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
3/4)), x_] :> Simp[-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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 229

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 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(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 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 312

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[Sqrt[(-b)*(x^2/a)]/(2*x)   Subst[Int[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 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[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 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]
 

3.4.22.4 Maple [F]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{4}}}{d \,x^{2}+c}d x\]

input

int((b*x^2+a)^(5/4)/(d*x^2+c),x)
 

output

int((b*x^2+a)^(5/4)/(d*x^2+c),x)
 

3.4.22.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\text {Timed out} \]

input

integrate((b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="fricas")
 

output

Timed out
 

3.4.22.6 Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{4}}}{c + d x^{2}}\, dx \]

input

integrate((b*x**2+a)**(5/4)/(d*x**2+c),x)
 

output

Integral((a + b*x**2)**(5/4)/(c + d*x**2), x)
 

3.4.22.7 Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{d x^{2} + c} \,d x } \]

input

integrate((b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="maxima")
 

output

integrate((b*x^2 + a)^(5/4)/(d*x^2 + c), x)
 

3.4.22.8 Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{d x^{2} + c} \,d x } \]

input

integrate((b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="giac")
 

output

integrate((b*x^2 + a)^(5/4)/(d*x^2 + c), x)
 

3.4.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/4}}{d\,x^2+c} \,d x \]

input

int((a + b*x^2)^(5/4)/(c + d*x^2),x)
 

output

int((a + b*x^2)^(5/4)/(c + d*x^2), x)