Integrand size = 32, antiderivative size = 318 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {2 (b B-a D) \sqrt {c x}}{b^2}+\frac {2 C (c x)^{3/2}}{3 b c}+\frac {2 D (c x)^{5/2}}{5 b c^2}-\frac {\sqrt {c} \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} b^{9/4}}+\frac {\sqrt {c} \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} b^{9/4}}-\frac {\sqrt {c} \left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} \sqrt [4]{a} b^{9/4}} \] Output:
2*(B*b-D*a)*(c*x)^(1/2)/b^2+2/3*C*(c*x)^(3/2)/b/c+2/5*D*(c*x)^(5/2)/b/c^2- 1/2*c^(1/2)*(b^(1/2)*(A*b-C*a)-a^(1/2)*(B*b-D*a))*arctan(1-2^(1/2)*b^(1/4) *(c*x)^(1/2)/a^(1/4)/c^(1/2))*2^(1/2)/a^(1/4)/b^(9/4)+1/2*c^(1/2)*(b^(1/2) *(A*b-C*a)-a^(1/2)*(B*b-D*a))*arctan(1+2^(1/2)*b^(1/4)*(c*x)^(1/2)/a^(1/4) /c^(1/2))*2^(1/2)/a^(1/4)/b^(9/4)-1/2*c^(1/2)*(b^(1/2)*(A*b-C*a)+a^(1/2)*( B*b-D*a))*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(c*x)^(1/2)/c^(1/2)/(a^(1/2)+b^( 1/2)*x))*2^(1/2)/a^(1/4)/b^(9/4)
Time = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {\sqrt {c x} \left (4 \sqrt [4]{b} (15 b B-15 a D+b x (5 C+3 D x))-\frac {15 \sqrt {2} \left (A b^{3/2}-\sqrt {a} b B-a \sqrt {b} C+a^{3/2} D\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a} \sqrt {x}}+\frac {15 \sqrt {2} \left (-A b^{3/2}-\sqrt {a} b B+a \sqrt {b} C+a^{3/2} D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a} \sqrt {x}}\right )}{30 b^{9/4}} \] Input:
Integrate[(Sqrt[c*x]*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
Output:
(Sqrt[c*x]*(4*b^(1/4)*(15*b*B - 15*a*D + b*x*(5*C + 3*D*x)) - (15*Sqrt[2]* (A*b^(3/2) - Sqrt[a]*b*B - a*Sqrt[b]*C + a^(3/2)*D)*ArcTan[(Sqrt[a] - Sqrt [b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(1/4)*Sqrt[x]) + (15*Sqrt[2] *(-(A*b^(3/2)) - Sqrt[a]*b*B + a*Sqrt[b]*C + a^(3/2)*D)*ArcTanh[(Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(1/4)*Sqrt[x])))/(30*b^ (9/4))
Time = 0.91 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2333, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (\frac {\sqrt {c x} (x (b B-a D)-a C+A b)}{b \left (a+b x^2\right )}+\frac {C \sqrt {c x}}{b}+\frac {D (c x)^{3/2}}{b c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ) \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right )}{\sqrt {2} \sqrt [4]{a} b^{9/4}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right ) \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right )}{\sqrt {2} \sqrt [4]{a} b^{9/4}}+\frac {\sqrt {c} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right )}{2 \sqrt {2} \sqrt [4]{a} b^{9/4}}-\frac {\sqrt {c} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right )}{2 \sqrt {2} \sqrt [4]{a} b^{9/4}}+\frac {2 \sqrt {c x} (b B-a D)}{b^2}+\frac {2 D (c x)^{5/2}}{5 b c^2}+\frac {2 C (c x)^{3/2}}{3 b c}\) |
Input:
Int[(Sqrt[c*x]*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2),x]
Output:
(2*(b*B - a*D)*Sqrt[c*x])/b^2 + (2*C*(c*x)^(3/2))/(3*b*c) + (2*D*(c*x)^(5/ 2))/(5*b*c^2) - (Sqrt[c]*(Sqrt[b]*(A*b - a*C) - Sqrt[a]*(b*B - a*D))*ArcTa n[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])])/(Sqrt[2]*a^(1/4)*b^( 9/4)) + (Sqrt[c]*(Sqrt[b]*(A*b - a*C) - Sqrt[a]*(b*B - a*D))*ArcTan[1 + (S qrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])])/(Sqrt[2]*a^(1/4)*b^(9/4)) + (Sqrt[c]*(Sqrt[b]*(A*b - a*C) + Sqrt[a]*(b*B - a*D))*Log[Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[c]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c*x]])/(2*Sqrt[2]*a^(1/4 )*b^(9/4)) - (Sqrt[c]*(Sqrt[b]*(A*b - a*C) + Sqrt[a]*(b*B - a*D))*Log[Sqrt [a]*Sqrt[c] + Sqrt[b]*Sqrt[c]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c*x]])/(2*S qrt[2]*a^(1/4)*b^(9/4))
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.76 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (\frac {D \left (c x \right )^{\frac {5}{2}} b}{5}+\frac {C c \left (c x \right )^{\frac {3}{2}} b}{3}+B b \,c^{2} \sqrt {c x}-D a \,c^{2} \sqrt {c x}\right )}{b^{2}}+\frac {2 c^{3} \left (\frac {\left (-a b B c +D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (b^{2} A -C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}}{c^{2}}\) | \(365\) |
| default | \(\frac {\frac {2 \left (\frac {D \left (c x \right )^{\frac {5}{2}} b}{5}+\frac {C c \left (c x \right )^{\frac {3}{2}} b}{3}+B b \,c^{2} \sqrt {c x}-D a \,c^{2} \sqrt {c x}\right )}{b^{2}}+\frac {2 c^{3} \left (\frac {\left (-a b B c +D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (b^{2} A -C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}}{c^{2}}\) | \(365\) |
| pseudoelliptic | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \left (B b -D a \right ) \sqrt {\frac {a \,c^{2}}{b}}}{4}+2 \sqrt {c x}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \left (\left (\frac {1}{5} D x^{2}+\frac {1}{3} C x +B \right ) b -D a \right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \left (A b -C a \right ) c}{4}}{b^{2} \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\) | \(373\) |
Input:
int((c*x)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
2/c^2*(1/b^2*(1/5*D*(c*x)^(5/2)*b+1/3*C*c*(c*x)^(3/2)*b+B*b*c^2*(c*x)^(1/2 )-D*a*c^2*(c*x)^(1/2))+c^3/b^2*(1/8*(-B*a*b*c+D*a^2*c)*(a*c^2/b)^(1/4)/a/c ^2*2^(1/2)*(ln((c*x+(a*c^2/b)^(1/4)*(c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2))/( c*x-(a*c^2/b)^(1/4)*(c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2)))+2*arctan(2^(1/2) /(a*c^2/b)^(1/4)*(c*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/ 2)-1))+1/8*(A*b^2-C*a*b)/b/(a*c^2/b)^(1/4)*2^(1/2)*(ln((c*x-(a*c^2/b)^(1/4 )*(c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2))/(c*x+(a*c^2/b)^(1/4)*(c*x)^(1/2)*2^ (1/2)+(a*c^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)+1)+2* arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1))))
Leaf count of result is larger than twice the leaf count of optimal. 3318 vs. \(2 (239) = 478\).
Time = 0.22 (sec) , antiderivative size = 3318, normalized size of antiderivative = 10.43 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\text {Too large to display} \] Input:
integrate((c*x)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 24.68 (sec) , antiderivative size = 1164, normalized size of antiderivative = 3.66 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\text {Too large to display} \] Input:
integrate((c*x)**(1/2)*(D*x**3+C*x**2+B*x+A)/(b*x**2+a),x)
Output:
-3*A*sqrt(c)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a** (1/4))*gamma(3/4)/(8*a**(1/4)*b**(3/4)*gamma(7/4)) - 3*I*A*sqrt(c)*exp(-3* I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(3/4)/ (8*a**(1/4)*b**(3/4)*gamma(7/4)) + 3*A*sqrt(c)*exp(-3*I*pi/4)*log(1 - b**( 1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(3/4)/(8*a**(1/4)*b**(3/4) *gamma(7/4)) + 3*I*A*sqrt(c)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_p olar(7*I*pi/4)/a**(1/4))*gamma(3/4)/(8*a**(1/4)*b**(3/4)*gamma(7/4)) + 5*B *a**(1/4)*sqrt(c)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/ a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) - 5*I*B*a**(1/4)*sqrt(c)*exp( -I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(5/4) /(8*b**(5/4)*gamma(9/4)) - 5*B*a**(1/4)*sqrt(c)*exp(-I*pi/4)*log(1 - b**(1 /4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4 )) + 5*I*B*a**(1/4)*sqrt(c)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_pola r(7*I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) + 5*B*sqrt(c)*sqr t(x)*gamma(5/4)/(2*b*gamma(9/4)) + 7*C*a**(3/4)*sqrt(c)*exp(-3*I*pi/4)*log (1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*g amma(11/4)) + 7*I*C*a**(3/4)*sqrt(c)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt( x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*gamma(11/4)) - 7*C *a**(3/4)*sqrt(c)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi /4)/a**(1/4))*gamma(7/4)/(8*b**(7/4)*gamma(11/4)) - 7*I*C*a**(3/4)*sqrt...
Time = 0.13 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {\frac {15 \, c^{2} {\left (\frac {\sqrt {2} {\left ({\left (C a b - A b^{2}\right )} \sqrt {a} c + {\left (D a^{2} \sqrt {b} - B a b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x + \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left ({\left (C a b - A b^{2}\right )} \sqrt {a} c + {\left (D a^{2} \sqrt {b} - B a b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x - \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {2 \, \sqrt {2} {\left ({\left (C a b - A b^{2}\right )} \sqrt {a} c - {\left (D a^{2} \sqrt {b} - B a b^{\frac {3}{2}}\right )} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c} - \frac {2 \, \sqrt {2} {\left ({\left (C a b - A b^{2}\right )} \sqrt {a} c - {\left (D a^{2} \sqrt {b} - B a b^{\frac {3}{2}}\right )} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c}\right )}}{b^{2}} + \frac {8 \, {\left (3 \, \left (c x\right )^{\frac {5}{2}} D b + 5 \, \left (c x\right )^{\frac {3}{2}} C b c - 15 \, {\left (D a - B b\right )} \sqrt {c x} c^{2}\right )}}{b^{2} c}}{60 \, c} \] Input:
integrate((c*x)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="maxima")
Output:
1/60*(15*c^2*(sqrt(2)*((C*a*b - A*b^2)*sqrt(a)*c + (D*a^2*sqrt(b) - B*a*b^ (3/2))*c)*log(sqrt(b)*c*x + sqrt(2)*(a*c^2)^(1/4)*sqrt(c*x)*b^(1/4) + sqrt (a)*c)/((a*c^2)^(3/4)*b^(3/4)) - sqrt(2)*((C*a*b - A*b^2)*sqrt(a)*c + (D*a ^2*sqrt(b) - B*a*b^(3/2))*c)*log(sqrt(b)*c*x - sqrt(2)*(a*c^2)^(1/4)*sqrt( c*x)*b^(1/4) + sqrt(a)*c)/((a*c^2)^(3/4)*b^(3/4)) - 2*sqrt(2)*((C*a*b - A* b^2)*sqrt(a)*c - (D*a^2*sqrt(b) - B*a*b^(3/2))*c)*arctan(1/2*sqrt(2)*(sqrt (2)*(a*c^2)^(1/4)*b^(1/4) + 2*sqrt(c*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/ (sqrt(sqrt(a)*sqrt(b)*c)*sqrt(a)*sqrt(b)*c) - 2*sqrt(2)*((C*a*b - A*b^2)*s qrt(a)*c - (D*a^2*sqrt(b) - B*a*b^(3/2))*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*( a*c^2)^(1/4)*b^(1/4) - 2*sqrt(c*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/(sqrt (sqrt(a)*sqrt(b)*c)*sqrt(a)*sqrt(b)*c))/b^2 + 8*(3*(c*x)^(5/2)*D*b + 5*(c* x)^(3/2)*C*b*c - 15*(D*a - B*b)*sqrt(c*x)*c^2)/(b^2*c))/c
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (239) = 478\).
Time = 0.14 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{4} c} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{4} c} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \log \left (c x + \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{4 \, a b^{4} c} - \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \log \left (c x - \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{4 \, a b^{4} c} + \frac {2 \, {\left (3 \, \sqrt {c x} D b^{4} c^{10} x^{2} + 5 \, \sqrt {c x} C b^{4} c^{10} x - 15 \, \sqrt {c x} D a b^{3} c^{10} + 15 \, \sqrt {c x} B b^{4} c^{10}\right )}}{15 \, b^{5} c^{10}} \] Input:
integrate((c*x)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*sqrt(2)*((a*b^3*c^2)^(1/4)*D*a^2*b*c - (a*b^3*c^2)^(1/4)*B*a*b^2*c - ( a*b^3*c^2)^(3/4)*C*a + (a*b^3*c^2)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)* (a*c^2/b)^(1/4) + 2*sqrt(c*x))/(a*c^2/b)^(1/4))/(a*b^4*c) + 1/2*sqrt(2)*(( a*b^3*c^2)^(1/4)*D*a^2*b*c - (a*b^3*c^2)^(1/4)*B*a*b^2*c - (a*b^3*c^2)^(3/ 4)*C*a + (a*b^3*c^2)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*c^2/b)^(1/ 4) - 2*sqrt(c*x))/(a*c^2/b)^(1/4))/(a*b^4*c) + 1/4*sqrt(2)*((a*b^3*c^2)^(1 /4)*D*a^2*b*c - (a*b^3*c^2)^(1/4)*B*a*b^2*c + (a*b^3*c^2)^(3/4)*C*a - (a*b ^3*c^2)^(3/4)*A*b)*log(c*x + sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sqrt(a*c^ 2/b))/(a*b^4*c) - 1/4*sqrt(2)*((a*b^3*c^2)^(1/4)*D*a^2*b*c - (a*b^3*c^2)^( 1/4)*B*a*b^2*c + (a*b^3*c^2)^(3/4)*C*a - (a*b^3*c^2)^(3/4)*A*b)*log(c*x - sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sqrt(a*c^2/b))/(a*b^4*c) + 2/15*(3*sqr t(c*x)*D*b^4*c^10*x^2 + 5*sqrt(c*x)*C*b^4*c^10*x - 15*sqrt(c*x)*D*a*b^3*c^ 10 + 15*sqrt(c*x)*B*b^4*c^10)/(b^5*c^10)
Timed out. \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx=\int \frac {\sqrt {c\,x}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{b\,x^2+a} \,d x \] Input:
int(((c*x)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2),x)
Output:
int(((c*x)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2), x)
Time = 0.16 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {c x} \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx =\text {Too large to display} \] Input:
int((c*x)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a),x)
Output:
(sqrt(c)*( - 30*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2 + 30*b**(1/4)*a**(3 /4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4) *a**(1/4)*sqrt(2)))*b*c - 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**( 1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*d + 30*b* *(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b ))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2 + 30*b**(1/4)*a**(3/4)*sqrt(2)*atan(( b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)) )*b**2 - 30*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2* sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c + 30*b**(3/4)*a**(1/4)*s qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 1/4)*sqrt(2)))*a*d - 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)* sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2 + 15*b**(1/ 4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + s qrt(b)*x)*b**2 - 15*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**( 1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c - 15*b**(1/4)*a**(3/4)*sqrt(2)*log (sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2 + 15*b**(1/ 4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt (b)*x)*b*c - 15*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4) *sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d + 15*b**(3/4)*a**(1/4)*sqrt(2)*log(...