\(\int (a+c x^2)^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} (d+e x+f x^2) \, dx\) [4]

Optimal result

Integrand size = 41, antiderivative size = 54 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=-\frac {d (a e-(c d-a f) x) \left (a+c x^2\right )^{\frac {1}{2} \left (-1+\frac {a f}{c d}\right )}}{a (c d-a f)} \] Output:

-d*(a*e-(-a*f+c*d)*x)*(c*x^2+a)^(-1/2+1/2*a*f/c/d)/a/(-a*f+c*d)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 6.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.87 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=\frac {\left (a+c x^2\right )^{\frac {1}{2} \left (-1+\frac {a f}{c d}\right )} \left (1+\frac {c x^2}{a}\right )^{-\frac {a f}{2 c d}} \left (3 a d e \left (1+\frac {c x^2}{a}\right )^{\frac {a f}{2 c d}}+3 d (-c d+a f) x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (3-\frac {a f}{c d}\right ),\frac {3}{2},-\frac {c x^2}{a}\right )+f (-c d+a f) x^3 \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2} \left (3-\frac {a f}{c d}\right ),\frac {5}{2},-\frac {c x^2}{a}\right )\right )}{3 a (-c d+a f)} \] Input:

Integrate[(a + c*x^2)^((-6*c^2*d + 2*a*c*f)/(4*c^2*d))*(d + e*x + f*x^2),x 
]
 

Output:

((a + c*x^2)^((-1 + (a*f)/(c*d))/2)*(3*a*d*e*(1 + (c*x^2)/a)^((a*f)/(2*c*d 
)) + 3*d*(-(c*d) + a*f)*x*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, (3 - 
(a*f)/(c*d))/2, 3/2, -((c*x^2)/a)] + f*(-(c*d) + a*f)*x^3*Sqrt[1 + (c*x^2) 
/a]*Hypergeometric2F1[3/2, (3 - (a*f)/(c*d))/2, 5/2, -((c*x^2)/a)]))/(3*a* 
(-(c*d) + a*f)*(1 + (c*x^2)/a)^((a*f)/(2*c*d)))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2346, 27, 241}

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.

\(\displaystyle \int \left (d+e x+f x^2\right ) \left (a+c x^2\right )^{\frac {2 a c f-6 c^2 d}{4 c^2 d}} \, dx\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {d \int \frac {a e f x \left (c x^2+a\right )^{\frac {1}{2} \left (\frac {a f}{c d}-3\right )}}{d}dx}{a f}+\frac {d x \left (a+c x^2\right )^{\frac {1}{2} \left (\frac {a f}{c d}-1\right )}}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle e \int x \left (c x^2+a\right )^{\frac {1}{2} \left (\frac {a f}{c d}-3\right )}dx+\frac {d x \left (a+c x^2\right )^{\frac {1}{2} \left (\frac {a f}{c d}-1\right )}}{a}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {d x \left (a+c x^2\right )^{\frac {1}{2} \left (\frac {a f}{c d}-1\right )}}{a}-\frac {d e \left (a+c x^2\right )^{\frac {1}{2} \left (\frac {a f}{c d}-1\right )}}{c d-a f}\)

Input:

Int[(a + c*x^2)^((-6*c^2*d + 2*a*c*f)/(4*c^2*d))*(d + e*x + f*x^2),x]
 

Output:

-((d*e*(a + c*x^2)^((-1 + (a*f)/(c*d))/2))/(c*d - a*f)) + (d*x*(a + c*x^2) 
^((-1 + (a*f)/(c*d))/2))/a
 

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 241

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

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02

method	result	size
gosper	\(\frac {d \left (c \,x^{2}+a \right )^{1+\frac {a f -3 c d}{2 c d}} \left (a f x -c d x +a e \right )}{a \left (a f -c d \right )}\)	\(55\)
orering	\(\frac {\left (a f x -c d x +a e \right ) d \left (c \,x^{2}+a \right ) \left (c \,x^{2}+a \right )^{\frac {2 a c f -6 c^{2} d}{4 c^{2} d}}}{a \left (a f -c d \right )}\)	\(64\)
risch	\(\frac {\left (a c f \,x^{3}-c^{2} d \,x^{3}+a c e \,x^{2}+f \,a^{2} x -a d x c +a^{2} e \right ) d \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}}}{a \left (a f -c d \right )}\)	\(81\)
norman	\(d x \,{\mathrm e}^{\frac {\left (2 a c f -6 c^{2} d \right ) \ln \left (c \,x^{2}+a \right )}{4 c^{2} d}}+\frac {a d e \,{\mathrm e}^{\frac {\left (2 a c f -6 c^{2} d \right ) \ln \left (c \,x^{2}+a \right )}{4 c^{2} d}}}{a f -c d}+\frac {c d \,x^{3} {\mathrm e}^{\frac {\left (2 a c f -6 c^{2} d \right ) \ln \left (c \,x^{2}+a \right )}{4 c^{2} d}}}{a}+\frac {d e c \,x^{2} {\mathrm e}^{\frac {\left (2 a c f -6 c^{2} d \right ) \ln \left (c \,x^{2}+a \right )}{4 c^{2} d}}}{a f -c d}\)	\(161\)
parallelrisch	\(\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}} a c d f -x^{3} \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}} c^{2} d^{2}+a c d e \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}} x^{2}+x \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}} a^{2} d f -x \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}} a c \,d^{2}+d e \,a^{2} \left (c \,x^{2}+a \right )^{\frac {a f -3 c d}{2 c d}}}{a \left (a f -c d \right )}\)	\(208\)

Input:

int((c*x^2+a)^(1/4*(2*a*c*f-6*c^2*d)/c^2/d)*(f*x^2+e*x+d),x,method=_RETURN 
VERBOSE)
 

Output:

d/a*(c*x^2+a)^(1+1/2/c*(a*f-3*c*d)/d)/(a*f-c*d)*(a*f*x-c*d*x+a*e)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.72 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=-\frac {a c d e x^{2} + a^{2} d e - {\left (c^{2} d^{2} - a c d f\right )} x^{3} - {\left (a c d^{2} - a^{2} d f\right )} x}{{\left (a c d - a^{2} f\right )} {\left (c x^{2} + a\right )}^{\frac {3 \, c d - a f}{2 \, c d}}} \] Input:

integrate((c*x^2+a)^(1/4*(2*a*c*f-6*c^2*d)/c^2/d)*(f*x^2+e*x+d),x, algorit 
hm="fricas")
 

Output:

-(a*c*d*e*x^2 + a^2*d*e - (c^2*d^2 - a*c*d*f)*x^3 - (a*c*d^2 - a^2*d*f)*x) 
/((a*c*d - a^2*f)*(c*x^2 + a)^(1/2*(3*c*d - a*f)/(c*d)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (41) = 82\).

Time = 21.71 (sec) , antiderivative size = 374, normalized size of antiderivative = 6.93 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=\begin {cases} \tilde {\infty } \left (d x + \frac {e x^{2}}{2} + \frac {f x^{3}}{3}\right ) & \text {for}\: a = 0 \wedge c = 0 \\e \left (\begin {cases} \tilde {\infty } x^{2} & \text {for}\: c = 0 \\- \frac {1}{c \sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {f x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge d = 0 \\- \frac {d x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} + e \left (\begin {cases} \tilde {\infty } x^{2} & \text {for}\: c = 0 \\- \frac {1}{c \sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {f x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {e \log {\left (x - \sqrt {- \frac {d}{f}} \right )}}{2 c} + \frac {e \log {\left (x + \sqrt {- \frac {d}{f}} \right )}}{2 c} + \frac {f x}{c} & \text {for}\: a = \frac {c d}{f} \\\frac {a^{2} d e \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} + \frac {a^{2} d f x \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} - \frac {a c d^{2} x \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} + \frac {a c d e x^{2} \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} + \frac {a c d f x^{3} \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} - \frac {c^{2} d^{2} x^{3} \left (a + c x^{2}\right )^{\frac {a f}{2 c d} - \frac {3}{2}}}{a^{2} f - a c d} & \text {otherwise} \end {cases} \] Input:

integrate((c*x**2+a)**(1/4*(2*a*c*f-6*c**2*d)/c**2/d)*(f*x**2+e*x+d),x)
 

Output:

Piecewise((zoo*(d*x + e*x**2/2 + f*x**3/3), Eq(a, 0) & Eq(c, 0)), (e*Piece 
wise((zoo*x**2, Eq(c, 0)), (-1/(c*sqrt(c*x**2)), True)) + f*x**3*log(x)/(c 
*x**2)**(3/2), Eq(a, 0) & Eq(d, 0)), (-d*x/(2*(c*x**2)**(3/2)) + e*Piecewi 
se((zoo*x**2, Eq(c, 0)), (-1/(c*sqrt(c*x**2)), True)) + f*x**3*log(x)/(c*x 
**2)**(3/2), Eq(a, 0)), (e*log(x - sqrt(-d/f))/(2*c) + e*log(x + sqrt(-d/f 
))/(2*c) + f*x/c, Eq(a, c*d/f)), (a**2*d*e*(a + c*x**2)**(a*f/(2*c*d) - 3/ 
2)/(a**2*f - a*c*d) + a**2*d*f*x*(a + c*x**2)**(a*f/(2*c*d) - 3/2)/(a**2*f 
 - a*c*d) - a*c*d**2*x*(a + c*x**2)**(a*f/(2*c*d) - 3/2)/(a**2*f - a*c*d) 
+ a*c*d*e*x**2*(a + c*x**2)**(a*f/(2*c*d) - 3/2)/(a**2*f - a*c*d) + a*c*d* 
f*x**3*(a + c*x**2)**(a*f/(2*c*d) - 3/2)/(a**2*f - a*c*d) - c**2*d**2*x**3 
*(a + c*x**2)**(a*f/(2*c*d) - 3/2)/(a**2*f - a*c*d), True))
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=-\frac {{\left (a d e - {\left (c d^{2} - a d f\right )} x\right )} {\left (c x^{2} + a\right )}^{\frac {a f}{2 \, c d}}}{{\left (a c d - a^{2} f\right )} \sqrt {c x^{2} + a}} \] Input:

integrate((c*x^2+a)^(1/4*(2*a*c*f-6*c^2*d)/c^2/d)*(f*x^2+e*x+d),x, algorit 
hm="maxima")
 

Output:

-(a*d*e - (c*d^2 - a*d*f)*x)*(c*x^2 + a)^(1/2*a*f/(c*d))/((a*c*d - a^2*f)* 
sqrt(c*x^2 + a))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (51) = 102\).

Time = 0.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.98 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=\frac {c^{2} d^{2} x^{3} e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )} - a c d f x^{3} e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )} - a c d e x^{2} e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )} + a c d^{2} x e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )} - a^{2} d f x e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )} - a^{2} d e e^{\left (-\frac {3 \, c d \log \left (c x^{2} + a\right ) - a f \log \left (c x^{2} + a\right )}{2 \, c d}\right )}}{a c d - a^{2} f} \] Input:

integrate((c*x^2+a)^(1/4*(2*a*c*f-6*c^2*d)/c^2/d)*(f*x^2+e*x+d),x, algorit 
hm="giac")
 

Output:

(c^2*d^2*x^3*e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)) - 
a*c*d*f*x^3*e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)) - a 
*c*d*e*x^2*e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)) + a* 
c*d^2*x*e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)) - a^2*d 
*f*x*e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)) - a^2*d*e* 
e^(-1/2*(3*c*d*log(c*x^2 + a) - a*f*log(c*x^2 + a))/(c*d)))/(a*c*d - a^2*f 
)
 

Mupad [B] (verification not implemented)

Time = 15.43 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.28 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=\frac {\frac {a^2\,d\,e}{a^2\,f-a\,c\,d}+\frac {a\,d\,x\,\left (a\,f-c\,d\right )}{a^2\,f-a\,c\,d}+\frac {c\,d\,x^3\,\left (a\,f-c\,d\right )}{a^2\,f-a\,c\,d}+\frac {a\,c\,d\,e\,x^2}{a^2\,f-a\,c\,d}}{{\left (c\,x^2+a\right )}^{\frac {\frac {3\,c^2\,d}{2}-\frac {a\,c\,f}{2}}{c^2\,d}}} \] Input:

int((d + e*x + f*x^2)/(a + c*x^2)^(((3*c^2*d)/2 - (a*c*f)/2)/(c^2*d)),x)
 

Output:

((a^2*d*e)/(a^2*f - a*c*d) + (a*d*x*(a*f - c*d))/(a^2*f - a*c*d) + (c*d*x^ 
3*(a*f - c*d))/(a^2*f - a*c*d) + (a*c*d*e*x^2)/(a^2*f - a*c*d))/(a + c*x^2 
)^(((3*c^2*d)/2 - (a*c*f)/2)/(c^2*d))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \left (a+c x^2\right )^{\frac {-6 c^2 d+2 a c f}{4 c^2 d}} \left (d+e x+f x^2\right ) \, dx=\frac {\left (c \,x^{2}+a \right )^{\frac {a f +c d}{2 c d}} d \left (a f x -c d x +a e \right )}{a \left (a c f \,x^{2}-c^{2} d \,x^{2}+a^{2} f -a c d \right )} \] Input:

int((c*x^2+a)^(1/4*(2*a*c*f-6*c^2*d)/c^2/d)*(f*x^2+e*x+d),x)
 

Output:

((a + c*x**2)**((a*f + c*d)/(2*c*d))*d*(a*e + a*f*x - c*d*x))/(a*(a**2*f - 
 a*c*d + a*c*f*x**2 - c**2*d*x**2))