Integrand size = 34, antiderivative size = 212 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\frac {4 c f \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{15 a d}+\frac {4 f \left (3 b-2 c^2+3 a x\right ) \sqrt {c+\sqrt {b+a x}}}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \] Output:
Unintegrable
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=-\frac {4 f \sqrt {c+\sqrt {b+a x}} \left (2 c^2-c \sqrt {b+a x}-3 (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \] Input:
Integrate[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]
Output:
(-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)
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.
\(\displaystyle \int \frac {\left (f x^2-g\right ) \sqrt {\sqrt {a x+b}+c}}{d x^2+e} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {2 \int -\frac {\sqrt {b+a x} \left (a^2 g-a^2 f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{d x^2+e}d\sqrt {b+a x}}{a^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \int \frac {\sqrt {b+a x} \left (a^2 g-a^2 f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{d x^2+e}d\sqrt {b+a x}}{a^3}\) |
\(\Big \downarrow \) 2091 |
\(\displaystyle -\frac {2 \int \frac {\sqrt {b+a x} \left (a^2 g-a^2 f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{\frac {d b^2}{a^2}-\frac {2 d (b+a x) b}{a^2}+\frac {d (b+a x)^2}{a^2}+e}d\sqrt {b+a x}}{a^3}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle -\frac {4 \int -\frac {(-b+c-a x) (b+a x) \left (a^2 g-f \left (b-(-b+c-a x)^2\right )^2\right )}{d (-b+c-a x)^4-2 b d (-b+c-a x)^2+b^2 d+a^2 e}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {4 \int \frac {(-b+c-a x) (b+a x) \left (a^2 g-f \left (b-(-b+c-a x)^2\right )^2\right )}{d (-b+c-a x)^4-2 b d (-b+c-a x)^2+b^2 d+a^2 e}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {4 \int \frac {(-b+c-a x) (b+a x) \left (a^2 g-f \left (b-(-b+c-a x)^2\right )^2\right )}{d (-b+c-a x)^4-2 b d (-b+c-a x)^2+b^2 d \left (\frac {e a^2}{b^2 d}+1\right )}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {4 \int \left (\frac {f (b+a x)^2}{d}-\frac {c f (b+a x)}{d}+\frac {\left (a^2 c (e f+d g)-a^2 (e f+d g) (b+a x)\right ) (b+a x)}{d \left (d (-b+c-a x)^4-2 b d (-b+c-a x)^2+b^2 d \left (\frac {e a^2}{b^2 d}+1\right )\right )}\right )d\sqrt {c+\sqrt {b+a x}}}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \left (-\frac {a^2 c (d g+e f) \int \frac {b+a x}{d (b+a x)^4-4 c d (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) d (b+a x)^2+4 b c \left (1-\frac {c^2}{b}\right ) d (b+a x)+b^2 d \left (\frac {d c^4-2 b d c^2+a^2 e}{b^2 d}+1\right )}d\sqrt {c+\sqrt {b+a x}}}{d}+\frac {a^2 (d g+e f) \int \frac {(b+a x)^2}{d (b+a x)^4-4 c d (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) d (b+a x)^2+4 b c \left (1-\frac {c^2}{b}\right ) d (b+a x)+b^2 d \left (\frac {d c^4-2 b d c^2+a^2 e}{b^2 d}+1\right )}d\sqrt {c+\sqrt {b+a x}}}{d}+\frac {c f (a x+b)^{3/2}}{3 d}-\frac {f (a x+b)^{5/2}}{5 d}\right )}{a}\) |
Input:
Int[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]
Output:
$Aborted
Time = 0.63 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {4 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\) | \(198\) |
default | \(-\frac {2 \left (\frac {2 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +e \,a^{2}+b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{4 d^{2}}\right )}{a}\) | \(198\) |
Input:
int((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x,method=_RETURNVERBOSE)
Output:
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* c*d*_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)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.59 (sec) , antiderivative size = 5974, normalized size of antiderivative = 28.18 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \] Input:
integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="fricas ")
Output:
Too large to include
Timed out. \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Timed out} \] Input:
integrate((f*x**2-g)*(c+(a*x+b)**(1/2))**(1/2)/(d*x**2+e),x)
Output:
Timed out
Not integrable
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int { \frac {{\left (f x^{2} - g\right )} \sqrt {c + \sqrt {a x + b}}}{d x^{2} + e} \,d x } \] Input:
integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="maxima ")
Output:
integrate((f*x^2 - g)*sqrt(c + sqrt(a*x + b))/(d*x^2 + e), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.91 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.45 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\text {Too large to display} \] Input:
integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="giac")
Output:
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 + 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 + s qrt((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 + sqr t(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*...
Not integrable
Time = 9.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int -\frac {\sqrt {c+\sqrt {b+a\,x}}\,\left (g-f\,x^2\right )}{d\,x^2+e} \,d x \] Input:
int(-((c + (b + a*x)^(1/2))^(1/2)*(g - f*x^2))/(e + d*x^2),x)
Output:
int(-((c + (b + a*x)^(1/2))^(1/2)*(g - f*x^2))/(e + d*x^2), x)
Not integrable
Time = 200.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx=\int \frac {\left (f \,x^{2}-g \right ) \sqrt {c +\sqrt {a x +b}}}{d \,x^{2}+e}d x \] Input:
int((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x)
Output:
int((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x)