\(\int \frac {(-g+f x^2) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx\) [2535]

Optimal result

Integrand size = 34, antiderivative size = 212 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\frac {4 c f \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{15 a d}+\frac {4 f \left (3 b-2 c^2+3 a x\right ) \sqrt {c+\sqrt {b+a x}}}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx \]

[In]

Int[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]

[Out]

(-4*c*f*(c + Sqrt[b + a*x])^(3/2))/(3*a*d) + (4*f*(c + Sqrt[b + a*x])^(5/2))/(5*a*d) + (4*a*(e*f + d*g)*Defer[
Subst][Defer[Int][x^4/(-(b^2*d*(1 + (-2*b*c^2*d + c^4*d + a^2*e)/(b^2*d))) - 4*b*c*(1 - c^2/b)*d*x^2 + 2*b*(1
- (3*c^2)/b)*d*x^4 + 4*c*d*x^6 - d*x^8), x], x, Sqrt[c + Sqrt[b + a*x]]])/d + (4*a*c*(e*f + d*g)*Defer[Subst][
Defer[Int][x^2/(b^2*d*(1 + (-2*b*c^2*d + c^4*d + a^2*e)/(b^2*d)) + 4*b*c*(1 - c^2/b)*d*x^2 - 2*b*(1 - (3*c^2)/
b)*d*x^4 - 4*c*d*x^6 + d*x^8), x], x, Sqrt[c + Sqrt[b + a*x]]])/d

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x \sqrt {c+x} \left (-a^2 g+f \left (b-x^2\right )^2\right )}{e+\frac {d \left (b-x^2\right )^2}{a^2}} \, dx,x,\sqrt {b+a x}\right )}{a^3} \\ & = \frac {4 \text {Subst}\left (\int \frac {x^2 \left (-c+x^2\right ) \left (-a^2 g+f \left (b-\left (c-x^2\right )^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3} \\ & = \frac {4 \text {Subst}\left (\int \left (-\frac {a^2 c f x^2}{d}+\frac {a^2 f x^4}{d}+\frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{d \left (e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \text {Subst}\left (\int \frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \text {Subst}\left (\int \frac {a^2 (e f+d g) x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \text {Subst}\left (\int \frac {x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \text {Subst}\left (\int \left (\frac {a^2 x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8}+\frac {a^2 c x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d} \\ & = -\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 a (e f+d g)) \text {Subst}\left (\int \frac {x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d}+\frac {(4 a c (e f+d g)) \text {Subst}\left (\int \frac {x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=-\frac {4 f \sqrt {c+\sqrt {b+a x}} \left (2 c^2-c \sqrt {b+a x}-3 (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \]

[In]

Integrate[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]

[Out]

(-4*f*Sqrt[c + Sqrt[b + a*x]]*(2*c^2 - c*Sqrt[b + a*x] - 3*(b + a*x)))/(15*a*d) - (a*(e*f + d*g)*RootSum[b^2*d
- 2*b*c^2*d + c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#1^8 &
, (Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1)/(-b + c^2 - 2*c*#1^2 + #1^4) & ])/(2*d^2)

Maple [N/A] (verified)

Time = 0.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {4 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 d c \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\)	\(198\)
default	\(-\frac {2 \left (\frac {2 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 d c \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{4 d^{2}}\right )}{a}\)	\(198\)

[In]

int((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x,method=_RETURNVERBOSE)

[Out]

2/a*(-2*f/d*(-1/5*(c+(a*x+b)^(1/2))^(5/2)+1/3*c*(c+(a*x+b)^(1/2))^(3/2))+1/4*a^2/d^2*sum(((-d*g-e*f)*_R^4+c*(d
*g+e*f)*_R^2)/(_R^7-3*_R^5*c+3*_R^3*c^2-_R^3*b-_R*c^3+_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(d*_Z^8-
4*d*c*_Z^6+(6*c^2*d-2*b*d)*_Z^4+(-4*c^3*d+4*b*c*d)*_Z^2+c^4*d-2*b*c^2*d+e*a^2+b^2*d)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.75 (sec) , antiderivative size = 5974, normalized size of antiderivative = 28.18 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \]

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int { \frac {{\left (f x^{2} - g\right )} \sqrt {c + \sqrt {a x + b}}}{d x^{2} + e} \,d x } \]

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="maxima")

[Out]

integrate((f*x^2 - g)*sqrt(c + sqrt(a*x + b))/(d*x^2 + e), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.44 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.45 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \]

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="giac")

[Out]

4/15*(3*a^4*(c + sqrt(a*x + b))^(5/2)*d^4*f - 5*a^4*(c + sqrt(a*x + b))^(3/2)*c*d^4*f)/(a^5*d^5) + 1/2*((a^7*(
c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c + sqrt
((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)
) + sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d)))/((c + sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + s
qrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c
+ sqrt((b*d + sqrt(-d*e)*a)/d))) - (a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c + sqrt((b*d + s
qrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d + sqrt(-d*e)*
a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d)))/((c + sqrt((b*d + sqrt(-
d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d + sqrt(-
d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d + sqrt(-d*e)
*a)/d))^2*d^4*e*f - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^
2*d^5*g - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c - sqrt((b*d + s
qrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d
+ (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d + sqrt(-d*e)*
a)/d))) - (a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*e*f
+ a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c
+ sqrt(a*x + b)) - sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c
- sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d
- b*c*d)*sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d))) + (a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c
+ sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b
*d - sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d)))/((c + sq
rt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sq
rt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d))) - (a^7*(c + sqrt((
b*d - sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c + sqrt((b*d - sq
rt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c
+ sqrt((b*d - sqrt(-d*e)*a)/d)))/((c + sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*
a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b
*d - sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*e*f - a^7*(c - sqrt((b*d - sqrt(-d*e)*
a)/d))*c*d^4*e*f + a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d
^5*g)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d)
)^(7/2)*d - 3*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d)
)^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d))) - (a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*
d^4*e*f - a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*e*f + a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g -
a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c - sqrt((b*d - sqrt(-d*e)*
a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*
d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d))))/(
a^6*d^5)

Mupad [N/A]

Not integrable

Time = 6.89 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int -\frac {\sqrt {c+\sqrt {b+a\,x}}\,\left (g-f\,x^2\right )}{d\,x^2+e} \,d x \]

[In]

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

[Out]

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