\(\int \frac {(d+e x)^3}{(a+b x^2+c x^4)^{3/2}} \, dx\) [266]

Optimal result

Integrand size = 24, antiderivative size = 481 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {d x \left (\left (b^2-2 a c\right ) d^2-3 a b e^2+c \left (b d^2-6 a e^2\right ) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {e \left (3 b d^2-2 a e^2+\left (6 c d^2-b e^2\right ) x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} d \left (b d^2-6 a e^2\right ) x \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{c} d \left (b d^2-6 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {d \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \sqrt {a+b x^2+c x^4}} \] Output:

d*x*((-2*a*c+b^2)*d^2-3*a*b*e^2+c*(-6*a*e^2+b*d^2)*x^2)/a/(-4*a*c+b^2)/(c* 
x^4+b*x^2+a)^(1/2)-e*(3*b*d^2-2*a*e^2+(-b*e^2+6*c*d^2)*x^2)/(-4*a*c+b^2)/( 
c*x^4+b*x^2+a)^(1/2)-c^(1/2)*d*(-6*a*e^2+b*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/a/ 
(-4*a*c+b^2)/(a^(1/2)+c^(1/2)*x^2)+c^(1/4)*d*(-6*a*e^2+b*d^2)*(a^(1/2)+c^( 
1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2* 
arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(-4*a* 
c+b^2)/(c*x^4+b*x^2+a)^(1/2)-1/2*d*(c^(1/2)*d^2-3*a^(1/2)*e^2)*(a^(1/2)+c^ 
(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM 
(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(b-2 
*a^(1/2)*c^(1/2))/c^(1/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.07 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (2 a^2 e^3+b d^3 x \left (b+c x^2\right )-a \left (b e \left (3 d^2+3 d e x-e^2 x^2\right )+2 c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) d \left (b d^2-6 a e^2\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i d \left (-b^2 d^2+4 a c d^2+b \sqrt {b^2-4 a c} d^2-6 a \sqrt {b^2-4 a c} e^2\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

-1/4*(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(2*a^2*e^3 + b*d^3*x*(b + c*x^2) 
- a*(b*e*(3*d^2 + 3*d*e*x - e^2*x^2) + 2*c*d*x*(d^2 + 3*d*e*x + 3*e^2*x^2) 
)) + I*(-b + Sqrt[b^2 - 4*a*c])*d*(b*d^2 - 6*a*e^2)*Sqrt[(b + Sqrt[b^2 - 4 
*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] 
+ 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b 
+ Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] 
 - I*d*(-(b^2*d^2) + 4*a*c*d^2 + b*Sqrt[b^2 - 4*a*c]*d^2 - 6*a*Sqrt[b^2 - 
4*a*c]*e^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]) 
]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Elli 
pticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 
- 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 
- 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2202, 1492, 27, 1511, 27, 1416, 1509, 1576, 27, 1158}

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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {\int \frac {d \left (c \left (b d^2-6 a e^2\right ) x^2+a \left (2 c d^2-3 b e^2\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \int \frac {c \left (b d^2-6 a e^2\right ) x^2+a \left (2 c d^2-3 b e^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {d \left (\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\sqrt {a} \sqrt {c} \left (b d^2-6 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx\right )}{a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \left (\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\sqrt {c} \left (b d^2-6 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )}{a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\sqrt {c} \left (b d^2-6 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )}{a \left (b^2-4 a c\right )}+\int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \int \frac {x \left (x^2 e^3+3 d^2 e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx-\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\sqrt {c} \left (b d^2-6 a e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )}{a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {1}{2} \int \frac {e \left (3 d^2+e^2 x^2\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2-\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\sqrt {c} \left (b d^2-6 a e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )}{a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} e \int \frac {3 d^2+e^2 x^2}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2-\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\sqrt {c} \left (b d^2-6 a e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )}{a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1158

\(\displaystyle -\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-3 \sqrt {a} e^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\sqrt {c} \left (b d^2-6 a e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )}{a \left (b^2-4 a c\right )}+\frac {d x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-6 a e^2\right )-3 a b e^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {e \left (-2 a e^2+x^2 \left (6 c d^2-b e^2\right )+3 b d^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

Input:

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

Output:

(d*x*((b^2 - 2*a*c)*d^2 - 3*a*b*e^2 + c*(b*d^2 - 6*a*e^2)*x^2))/(a*(b^2 - 
4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (e*(3*b*d^2 - 2*a*e^2 + (6*c*d^2 - b*e^2 
)*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (d*(-(Sqrt[c]*(b*d^2 - 6 
*a*e^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4) 
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^ 
2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/ 
(c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(Sq 
rt[c]*d^2 - 3*Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4 
)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - 
 b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/(a*(b^2 - 
4*a*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 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*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] && 
 LtQ[p, -1] && 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]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(-\frac {2 c \left (-\frac {d \left (6 a \,e^{2}-b \,d^{2}\right ) x^{3}}{2 a \left (4 a c -b^{2}\right )}+\frac {e \left (b \,e^{2}-6 c \,d^{2}\right ) x^{2}}{2 c \left (4 a c -b^{2}\right )}-\frac {d \left (3 a b \,e^{2}+2 a c \,d^{2}-b^{2} d^{2}\right ) x}{2 a c \left (4 a c -b^{2}\right )}+\frac {e \left (2 a \,e^{2}-3 b \,d^{2}\right )}{2 \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {d^{3}}{a}-\frac {d \left (3 a b \,e^{2}+2 a c \,d^{2}-b^{2} d^{2}\right )}{\left (4 a c -b^{2}\right ) a}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {c d \left (6 a \,e^{2}-b \,d^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(603\)
default	\(\text {Expression too large to display}\)	\(1017\)

Input:

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

Output:

-2*c*(-1/2*d*(6*a*e^2-b*d^2)/a/(4*a*c-b^2)*x^3+1/2*e*(b*e^2-6*c*d^2)/c/(4* 
a*c-b^2)*x^2-1/2*d*(3*a*b*e^2+2*a*c*d^2-b^2*d^2)/a/c/(4*a*c-b^2)*x+1/2*e*( 
2*a*e^2-3*b*d^2)/(4*a*c-b^2)/c)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/4*(d^3/a-d*( 
3*a*b*e^2+2*a*c*d^2-b^2*d^2)/(4*a*c-b^2)/a)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2 
))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2) 
^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(- 
4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+ 
1/2*c*d*(6*a*e^2-b*d^2)/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1 
/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/ 
a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x 
*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/ 
2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2), 
1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(sqrt(1/2)*(a*b^2*c*d^3 - 6*a^2*b*c*d*e^2 + (b^2*c^2*d^3 - 6*a*b*c^2* 
d*e^2)*x^4 + (b^3*c*d^3 - 6*a*b^2*c*d*e^2)*x^2 - (a^2*b*c*d^3 - 6*a^3*c*d* 
e^2 + (a*b*c^2*d^3 - 6*a^2*c^2*d*e^2)*x^4 + (a*b^2*c*d^3 - 6*a^2*b*c*d*e^2 
)*x^2)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - 
b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a 
)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*((2 
*a^2*b + a*b^2)*c*d^3 + ((2*a*b + b^2)*c^2*d^3 - 3*(a*b^2*c + 2*a*b*c^2)*d 
*e^2)*x^4 - 3*(a^2*b^2 + 2*a^2*b*c)*d*e^2 + ((2*a*b^2 + b^3)*c*d^3 - 3*(a* 
b^3 + 2*a*b^2*c)*d*e^2)*x^2 + ((2*a^3 - a^2*b)*c*d^3 + ((2*a^2 - a*b)*c^2* 
d^3 - 3*(a^2*b*c - 2*a^2*c^2)*d*e^2)*x^4 - 3*(a^3*b - 2*a^3*c)*d*e^2 + ((2 
*a^2*b - a*b^2)*c*d^3 - 3*(a^2*b^2 - 2*a^2*b*c)*d*e^2)*x^2)*sqrt((b^2 - 4* 
a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_f(arcs 
in(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^ 
2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) + 2*(3*a^2*b*c*d^2*e - 2*a^3*c*e^3 - 
 (a*b*c^2*d^3 - 6*a^2*c^2*d*e^2)*x^3 + (6*a^2*c^2*d^2*e - a^2*b*c*e^3)*x^2 
 + (3*a^2*b*c*d*e^2 - (a*b^2*c - 2*a^2*c^2)*d^3)*x)*sqrt(c*x^4 + b*x^2 + a 
))/(a^3*b^2*c - 4*a^4*c^2 + (a^2*b^2*c^2 - 4*a^3*c^3)*x^4 + (a^2*b^3*c - 4 
*a^3*b*c^2)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(48*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + c*x**4)/(a**2 
+ 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a**3*b* 
c**2*d**3 + 96*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + c*x 
**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8) 
,x)*a**3*c**3*d**3*x**2 - 8*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + 
 b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 
 + c**2*x**8),x)*a**2*b**3*c*d**3 + 96*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*i 
nt(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 
 2*b*c*x**6 + c**2*x**8),x)*a**2*b**2*c**2*d**3*x**2 + 336*sqrt(c)*sqrt(a 
+ b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a* 
c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a**2*b*c**3*d**3*x**4 + 22 
4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 
2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*a**2*c**4 
*d**3*x**6 - sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + c*x** 
4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x 
)*a*b**5*d**3 - 26*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt(a + b*x**2 + 
 c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x 
**8),x)*a*b**4*c*d**3*x**2 - 8*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*int(sqrt( 
a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x 
**6 + c**2*x**8),x)*a*b**3*c**2*d**3*x**4 + 208*sqrt(c)*sqrt(a + b*x**2...