\(\int (a c+(b c+a d) x^2+b d x^4)^{5/2} \, dx\) [12]

Optimal result

Integrand size = 25, antiderivative size = 629 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2} \, dx=\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 c^3 d+234 a^2 b^2 c^2 d^2-61 a^3 b c d^3+8 a^4 d^4\right ) x \left (c+d x^2\right )}{693 b^2 d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {x \left (4 b^4 c^4-11 a b^3 c^3 d-210 a^2 b^2 c^2 d^2-11 a^3 b c d^3+4 a^4 d^4-12 b d (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x^2\right ) \sqrt {a c+(b c+a d) x^2+b d x^4}}{693 b^2 d^2}+\frac {5 x \left (3 \left (6 a b c d+(b c+a d)^2\right )+7 b d (b c+a d) x^2\right ) \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}}{693 b d}+\frac {1}{11} x \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2}-\frac {\sqrt {a} (b c+a d) \left (8 b^4 c^4-61 a b^3 c^3 d+234 a^2 b^2 c^2 d^2-61 a^3 b c d^3+8 a^4 d^4\right ) \left (c+d x^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{693 b^{5/2} d^3 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {2 a^{3/2} \left (2 b^4 c^4-13 a b^3 c^3 d+150 a^2 b^2 c^2 d^2-13 a^3 b c d^3+2 a^4 d^4\right ) \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{693 b^{5/2} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

1/693*(a*d+b*c)*(8*a^4*d^4-61*a^3*b*c*d^3+234*a^2*b^2*c^2*d^2-61*a*b^3*c^3 
*d+8*b^4*c^4)*x*(d*x^2+c)/b^2/d^3/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/693* 
x*(4*b^4*c^4-11*a*b^3*c^3*d-210*a^2*b^2*c^2*d^2-11*a^3*b*c*d^3+4*a^4*d^4-1 
2*b*d*(a*d+b*c)*(8*a*b*c*d-(a*d+b*c)^2)*x^2)*(a*c+(a*d+b*c)*x^2+b*d*x^4)^( 
1/2)/b^2/d^2+5/693*x*(18*a*b*c*d+3*(a*d+b*c)^2+7*b*d*(a*d+b*c)*x^2)*(a*c+( 
a*d+b*c)*x^2+b*d*x^4)^(3/2)/b/d+1/11*x*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(5/2)-1 
/693*a^(1/2)*(a*d+b*c)*(8*a^4*d^4-61*a^3*b*c*d^3+234*a^2*b^2*c^2*d^2-61*a* 
b^3*c^3*d+8*b^4*c^4)*(d*x^2+c)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/ 
2),(1-a*d/b/c)^(1/2))/b^(5/2)/d^3/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(a*c+(a* 
d+b*c)*x^2+b*d*x^4)^(1/2)+2/693*a^(3/2)*(2*a^4*d^4-13*a^3*b*c*d^3+150*a^2* 
b^2*c^2*d^2-13*a*b^3*c^3*d+2*b^4*c^4)*(d*x^2+c)*InverseJacobiAM(arctan(b^( 
1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/d^2/(a*(d*x^2+c)/c/(b*x^2+a))^( 
1/2)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.77 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.72 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 a^4 d^4+a^3 b d^3 \left (26 c+3 d x^2\right )+a^2 b^2 d^2 \left (393 c^2+356 c d x^2+113 d^2 x^4\right )+a b^3 d \left (26 c^3+356 c^2 d x^2+442 c d^2 x^4+161 d^3 x^6\right )+b^4 \left (-4 c^4+3 c^3 d x^2+113 c^2 d^2 x^4+161 c d^3 x^6+63 d^4 x^8\right )\right )-i c \left (8 b^5 c^5-53 a b^4 c^4 d+173 a^2 b^3 c^3 d^2+173 a^3 b^2 c^2 d^3-53 a^4 b c d^4+8 a^5 d^5\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (8 b^5 c^5-57 a b^4 c^4 d+199 a^2 b^3 c^3 d^2-127 a^3 b^2 c^2 d^3-27 a^4 b c d^4+4 a^5 d^5\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{693 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*a^4*d^4 + a^3*b*d^3*(26*c + 3*d 
*x^2) + a^2*b^2*d^2*(393*c^2 + 356*c*d*x^2 + 113*d^2*x^4) + a*b^3*d*(26*c^ 
3 + 356*c^2*d*x^2 + 442*c*d^2*x^4 + 161*d^3*x^6) + b^4*(-4*c^4 + 3*c^3*d*x 
^2 + 113*c^2*d^2*x^4 + 161*c*d^3*x^6 + 63*d^4*x^8)) - I*c*(8*b^5*c^5 - 53* 
a*b^4*c^4*d + 173*a^2*b^3*c^3*d^2 + 173*a^3*b^2*c^2*d^3 - 53*a^4*b*c*d^4 + 
 8*a^5*d^5)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sq 
rt[b/a]*x], (a*d)/(b*c)] + I*c*(8*b^5*c^5 - 57*a*b^4*c^4*d + 199*a^2*b^3*c 
^3*d^2 - 127*a^3*b^2*c^2*d^3 - 27*a^4*b*c*d^4 + 4*a^5*d^5)*Sqrt[1 + (b*x^2 
)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/( 
693*a^2*(b/a)^(5/2)*d^3*Sqrt[(a + b*x^2)*(c + d*x^2)])
 

Rubi [A] (verified)

Time = 1.97 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1404, 1490, 1490, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1404

\(\displaystyle \frac {5}{11} \int \left ((b c+a d) x^2+2 a c\right ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}dx+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {5}{11} \left (\frac {\int \left (4 (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x^2+a c \left (36 a b c d-(b c+a d)^2\right )\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}dx}{21 b d}+\frac {x \left (7 b d x^2 (a d+b c)+3 \left ((a d+b c)^2+6 a b c d\right )\right ) \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{63 b d}\right )+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {5}{11} \left (\frac {\frac {\int \frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) x^2+2 a c \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{15 b d}-\frac {x \left (4 a^4 d^4-11 a^3 b c d^3-210 a^2 b^2 c^2 d^2-11 a b^3 c^3 d-12 b d x^2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )+4 b^4 c^4\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{15 b d}}{21 b d}+\frac {x \left (7 b d x^2 (a d+b c)+3 \left ((a d+b c)^2+6 a b c d\right )\right ) \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{63 b d}\right )+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {5}{11} \left (\frac {\frac {\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c} \left (2 a^4 d^4-13 a^3 b c d^3+150 a^2 b^2 c^2 d^2-13 a b^3 c^3 d+2 b^4 c^4\right )+\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {\sqrt {a} \sqrt {c} (a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (4 a^4 d^4-11 a^3 b c d^3-210 a^2 b^2 c^2 d^2-11 a b^3 c^3 d-12 b d x^2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )+4 b^4 c^4\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{15 b d}}{21 b d}+\frac {x \left (7 b d x^2 (a d+b c)+3 \left ((a d+b c)^2+6 a b c d\right )\right ) \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{63 b d}\right )+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{11} \left (\frac {\frac {\sqrt {a} \sqrt {c} \left (2 \sqrt {a} \sqrt {c} \left (2 a^4 d^4-13 a^3 b c d^3+150 a^2 b^2 c^2 d^2-13 a b^3 c^3 d+2 b^4 c^4\right )+\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )}{\sqrt {b} \sqrt {d}}\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx-\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (4 a^4 d^4-11 a^3 b c d^3-210 a^2 b^2 c^2 d^2-11 a b^3 c^3 d-12 b d x^2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )+4 b^4 c^4\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{15 b d}}{21 b d}+\frac {x \left (7 b d x^2 (a d+b c)+3 \left ((a d+b c)^2+6 a b c d\right )\right ) \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{63 b d}\right )+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {5}{11} \left (\frac {\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c} \left (2 a^4 d^4-13 a^3 b c d^3+150 a^2 b^2 c^2 d^2-13 a b^3 c^3 d+2 b^4 c^4\right )+\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right )}{\sqrt {b} \sqrt {d}}\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {(a d+b c) \left (8 a^4 d^4-61 a^3 b c d^3+234 a^2 b^2 c^2 d^2-61 a b^3 c^3 d+8 b^4 c^4\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (4 a^4 d^4-11 a^3 b c d^3-210 a^2 b^2 c^2 d^2-11 a b^3 c^3 d-12 b d x^2 (a d+b c) \left (8 a b c d-(a d+b c)^2\right )+4 b^4 c^4\right ) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{15 b d}}{21 b d}+\frac {x \left (7 b d x^2 (a d+b c)+3 \left ((a d+b c)^2+6 a b c d\right )\right ) \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{63 b d}\right )+\frac {1}{11} x \left (x^2 (a d+b c)+a c+b d x^4\right )^{5/2}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{11} x \left (b d x^4+(b c+a d) x^2+a c\right )^{5/2}+\frac {5}{11} \left (\frac {x \left (7 b d (b c+a d) x^2+3 \left ((b c+a d)^2+6 a b c d\right )\right ) \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{63 b d}+\frac {\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c} \left (2 b^4 c^4-13 a b^3 d c^3+150 a^2 b^2 d^2 c^2-13 a^3 b d^3 c+2 a^4 d^4\right )+\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right )}{\sqrt {b} \sqrt {d}}\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 \sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {(b c+a d) \left (8 b^4 c^4-61 a b^3 d c^3+234 a^2 b^2 d^2 c^2-61 a^3 b d^3 c+8 a^4 d^4\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {x \sqrt {b d x^4+(b c+a d) x^2+a c}}{\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}}\right )}{\sqrt {b} \sqrt {d}}}{15 b d}-\frac {x \left (4 b^4 c^4-11 a b^3 d c^3-210 a^2 b^2 d^2 c^2-11 a^3 b d^3 c+4 a^4 d^4-12 b d (b c+a d) \left (8 a b c d-(b c+a d)^2\right ) x^2\right ) \sqrt {b d x^4+(b c+a d) x^2+a c}}{15 b d}}{21 b d}\right )\)

Input:

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

Output:

(x*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(5/2))/11 + (5*((x*(3*(6*a*b*c*d + (b 
*c + a*d)^2) + 7*b*d*(b*c + a*d)*x^2)*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(3 
/2))/(63*b*d) + (-1/15*(x*(4*b^4*c^4 - 11*a*b^3*c^3*d - 210*a^2*b^2*c^2*d^ 
2 - 11*a^3*b*c*d^3 + 4*a^4*d^4 - 12*b*d*(b*c + a*d)*(8*a*b*c*d - (b*c + a* 
d)^2)*x^2)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(b*d) + (-(((b*c + a*d)* 
(8*b^4*c^4 - 61*a*b^3*c^3*d + 234*a^2*b^2*c^2*d^2 - 61*a^3*b*c*d^3 + 8*a^4 
*d^4)*(-((x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt 
[b]*Sqrt[d]*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^ 
2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[ 
d]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - 
 (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[ 
a*c + (b*c + a*d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d])) + (a^(1/4)*c^(1/4)* 
(2*Sqrt[a]*Sqrt[c]*(2*b^4*c^4 - 13*a*b^3*c^3*d + 150*a^2*b^2*c^2*d^2 - 13* 
a^3*b*c*d^3 + 2*a^4*d^4) + ((b*c + a*d)*(8*b^4*c^4 - 61*a*b^3*c^3*d + 234* 
a^2*b^2*c^2*d^2 - 61*a^3*b*c*d^3 + 8*a^4*d^4))/(Sqrt[b]*Sqrt[d]))*(Sqrt[a] 
*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sq 
rt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4 
)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d] 
))/4])/(2*b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]))/(15*b*d) 
)/(21*b*d)))/11
 

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 1404

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

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.43

method	result	size
risch	\(-\frac {x \left (-63 b^{4} d^{4} x^{8}-161 a \,b^{3} d^{4} x^{6}-161 b^{4} c \,d^{3} x^{6}-113 a^{2} b^{2} d^{4} x^{4}-442 a \,b^{3} c \,d^{3} x^{4}-113 b^{4} c^{2} d^{2} x^{4}-3 a^{3} b \,d^{4} x^{2}-356 a^{2} b^{2} c \,d^{3} x^{2}-356 a \,b^{3} c^{2} d^{2} x^{2}-3 b^{4} c^{3} d \,x^{2}+4 a^{4} d^{4}-26 a^{3} b c \,d^{3}-393 a^{2} b^{2} c^{2} d^{2}-26 a \,b^{3} c^{3} d +4 b^{4} c^{4}\right ) \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{693 b^{2} d^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}+\frac {-\frac {\left (8 a^{5} d^{5}-53 a^{4} b c \,d^{4}+173 a^{3} b^{2} c^{2} d^{3}+173 a^{2} b^{3} c^{3} d^{2}-53 a \,b^{4} c^{4} d +8 b^{5} c^{5}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {4 a \,b^{4} c^{5} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {4 a^{5} c \,d^{4} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {26 a^{2} b^{3} c^{4} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {300 a^{3} b^{2} c^{3} d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {26 a^{4} b \,c^{2} d^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}}{693 b^{2} d^{2}}\)	\(901\)
default	\(\text {Expression too large to display}\)	\(1880\)
elliptic	\(\text {Expression too large to display}\)	\(1880\)

Input:

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

Output:

-1/693/b^2/d^2*x*(-63*b^4*d^4*x^8-161*a*b^3*d^4*x^6-161*b^4*c*d^3*x^6-113* 
a^2*b^2*d^4*x^4-442*a*b^3*c*d^3*x^4-113*b^4*c^2*d^2*x^4-3*a^3*b*d^4*x^2-35 
6*a^2*b^2*c*d^3*x^2-356*a*b^3*c^2*d^2*x^2-3*b^4*c^3*d*x^2+4*a^4*d^4-26*a^3 
*b*c*d^3-393*a^2*b^2*c^2*d^2-26*a*b^3*c^3*d+4*b^4*c^4)*(b*x^2+a)*(d*x^2+c) 
/((b*x^2+a)*(d*x^2+c))^(1/2)+1/693/b^2/d^2*(-(8*a^5*d^5-53*a^4*b*c*d^4+173 
*a^3*b^2*c^2*d^3+173*a^2*b^3*c^3*d^2-53*a*b^4*c^4*d+8*b^5*c^5)*c/(-b/a)^(1 
/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2 
)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a) 
^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+4*a*b^4*c^5/(-b/a)^(1/2)*(1+b*x^2/a)^(1/ 
2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a 
)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+4*a^5*c*d^4/(-b/a)^(1/2)*(1+b*x^2/a)^(1/ 
2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a 
)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-26*a^2*b^3*c^4*d/(-b/a)^(1/2)*(1+b*x^2/a 
)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x* 
(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+300*a^3*b^2*c^3*d^2/(-b/a)^(1/2)*(1 
+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*Elli 
pticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-26*a^4*b*c^2*d^3/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.87 \[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2} \, dx=-\frac {{\left (8 \, b^{5} c^{6} - 53 \, a b^{4} c^{5} d + 173 \, a^{2} b^{3} c^{4} d^{2} + 173 \, a^{3} b^{2} c^{3} d^{3} - 53 \, a^{4} b c^{2} d^{4} + 8 \, a^{5} c d^{5}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{5} c^{6} - 53 \, a b^{4} c^{5} d + 4 \, a^{5} d^{6} + {\left (173 \, a^{2} b^{3} + 4 \, a b^{4}\right )} c^{4} d^{2} + {\left (173 \, a^{3} b^{2} - 26 \, a^{2} b^{3}\right )} c^{3} d^{3} - {\left (53 \, a^{4} b - 300 \, a^{3} b^{2}\right )} c^{2} d^{4} + 2 \, {\left (4 \, a^{5} - 13 \, a^{4} b\right )} c d^{5}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (63 \, b^{5} d^{6} x^{10} + 8 \, b^{5} c^{5} d - 53 \, a b^{4} c^{4} d^{2} + 173 \, a^{2} b^{3} c^{3} d^{3} + 173 \, a^{3} b^{2} c^{2} d^{4} - 53 \, a^{4} b c d^{5} + 8 \, a^{5} d^{6} + 161 \, {\left (b^{5} c d^{5} + a b^{4} d^{6}\right )} x^{8} + {\left (113 \, b^{5} c^{2} d^{4} + 442 \, a b^{4} c d^{5} + 113 \, a^{2} b^{3} d^{6}\right )} x^{6} + {\left (3 \, b^{5} c^{3} d^{3} + 356 \, a b^{4} c^{2} d^{4} + 356 \, a^{2} b^{3} c d^{5} + 3 \, a^{3} b^{2} d^{6}\right )} x^{4} - {\left (4 \, b^{5} c^{4} d^{2} - 26 \, a b^{4} c^{3} d^{3} - 393 \, a^{2} b^{3} c^{2} d^{4} - 26 \, a^{3} b^{2} c d^{5} + 4 \, a^{4} b d^{6}\right )} x^{2}\right )} \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}}{693 \, b^{3} d^{4} x} \] Input:

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

Output:

-1/693*((8*b^5*c^6 - 53*a*b^4*c^5*d + 173*a^2*b^3*c^4*d^2 + 173*a^3*b^2*c^ 
3*d^3 - 53*a^4*b*c^2*d^4 + 8*a^5*c*d^5)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e( 
arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (8*b^5*c^6 - 53*a*b^4*c^5*d + 4*a^5*d^6 
 + (173*a^2*b^3 + 4*a*b^4)*c^4*d^2 + (173*a^3*b^2 - 26*a^2*b^3)*c^3*d^3 - 
(53*a^4*b - 300*a^3*b^2)*c^2*d^4 + 2*(4*a^5 - 13*a^4*b)*c*d^5)*sqrt(b*d)*x 
*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (63*b^5*d^6*x^10 
 + 8*b^5*c^5*d - 53*a*b^4*c^4*d^2 + 173*a^2*b^3*c^3*d^3 + 173*a^3*b^2*c^2* 
d^4 - 53*a^4*b*c*d^5 + 8*a^5*d^6 + 161*(b^5*c*d^5 + a*b^4*d^6)*x^8 + (113* 
b^5*c^2*d^4 + 442*a*b^4*c*d^5 + 113*a^2*b^3*d^6)*x^6 + (3*b^5*c^3*d^3 + 35 
6*a*b^4*c^2*d^4 + 356*a^2*b^3*c*d^5 + 3*a^3*b^2*d^6)*x^4 - (4*b^5*c^4*d^2 
- 26*a*b^4*c^3*d^3 - 393*a^2*b^3*c^2*d^4 - 26*a^3*b^2*c*d^5 + 4*a^4*b*d^6) 
*x^2)*sqrt(b*d*x^4 + (b*c + a*d)*x^2 + a*c))/(b^3*d^4*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a c+(b c+a d) x^2+b d x^4\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4*d**4*x + 26*sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*a**3*b*c*d**3*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a** 
3*b*d**4*x**3 + 393*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c**2*d**2* 
x + 356*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c*d**3*x**3 + 113*sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*d**4*x**5 + 26*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b**3*c**3*d*x + 356*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b 
**3*c**2*d**2*x**3 + 442*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**3*x 
**5 + 161*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*d**4*x**7 - 4*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*b**4*c**4*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*b**4*c**3*d*x**3 + 113*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c**2*d**2*x 
**5 + 161*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c*d**3*x**7 + 63*sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*b**4*d**4*x**9 + 8*int((sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**5*d**5 - 53*i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b 
*d*x**4),x)*a**4*b*c*d**4 + 173*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x** 
2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*b**2*c**2*d**3 + 173*int 
((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d 
*x**4),x)*a**2*b**3*c**3*d**2 - 53*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**4*c**4*d + 8*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x...