Integrand size = 32, antiderivative size = 295 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 A}{3 a c (c x)^{3/2}}-\frac {2 B}{a c^2 \sqrt {c x}}+\frac {\left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} a^{7/4} b^{3/4} c^{5/2}}-\frac {\left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} a^{7/4} b^{3/4} c^{5/2}}-\frac {\left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{7/4} b^{3/4} c^{5/2}} \] Output:
-2/3*A/a/c/(c*x)^(3/2)-2*B/a/c^2/(c*x)^(1/2)+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^(7/4)/b^(3/4)/c^(5/2)-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^(7/4)/b^(3/4)/c^( 5/2)-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^(7/4)/b^(3/4)/c^(5 /2)
Time = 0.39 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-4 a^{3/4} b^{3/4} (A+3 B x)+3 \sqrt {2} \left (A b^{3/2}+\sqrt {a} b B-a \sqrt {b} C-a^{3/2} D\right ) x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} \left (A b^{3/2}-\sqrt {a} b B-a \sqrt {b} C+a^{3/2} D\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{6 a^{7/4} b^{3/4} (c x)^{5/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/((c*x)^(5/2)*(a + b*x^2)),x]
Output:
(x*(-4*a^(3/4)*b^(3/4)*(A + 3*B*x) + 3*Sqrt[2]*(A*b^(3/2) + Sqrt[a]*b*B - a*Sqrt[b]*C - a^(3/2)*D)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^( 1/4)*b^(1/4)*Sqrt[x])] - 3*Sqrt[2]*(A*b^(3/2) - Sqrt[a]*b*B - a*Sqrt[b]*C + a^(3/2)*D)*x^(3/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(6*a^(7/4)*b^(3/4)*(c*x)^(5/2))
Time = 1.02 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.55, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (\frac {x (b B-a D)-a C+A b}{b (c x)^{5/2} \left (a+b x^2\right )}+\frac {C}{b (c x)^{5/2}}+\frac {D}{b c (c x)^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ) \left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right )}{\sqrt {2} a^{7/4} b^{3/4} c^{5/2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right ) \left (\sqrt {b} (A b-a C)+\sqrt {a} (b B-a D)\right )}{\sqrt {2} a^{7/4} b^{3/4} c^{5/2}}+\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right )}{2 \sqrt {2} a^{7/4} b^{3/4} c^{5/2}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (A b-a C)-\sqrt {a} (b B-a D)\right )}{2 \sqrt {2} a^{7/4} b^{3/4} c^{5/2}}-\frac {2 (A b-a C)}{3 a b c (c x)^{3/2}}-\frac {2 (b B-a D)}{a b c^2 \sqrt {c x}}-\frac {2 D}{b c^2 \sqrt {c x}}-\frac {2 C}{3 b c (c x)^{3/2}}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/((c*x)^(5/2)*(a + b*x^2)),x]
Output:
(-2*C)/(3*b*c*(c*x)^(3/2)) - (2*(A*b - a*C))/(3*a*b*c*(c*x)^(3/2)) - (2*D) /(b*c^2*Sqrt[c*x]) - (2*(b*B - a*D))/(a*b*c^2*Sqrt[c*x]) + ((Sqrt[b]*(A*b - a*C) + Sqrt[a]*(b*B - a*D))*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1 /4)*Sqrt[c])])/(Sqrt[2]*a^(7/4)*b^(3/4)*c^(5/2)) - ((Sqrt[b]*(A*b - a*C) + Sqrt[a]*(b*B - a*D))*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt [c])])/(Sqrt[2]*a^(7/4)*b^(3/4)*c^(5/2)) + ((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^(7/4)*b^(3/4)*c^(5/2)) - ((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^(7/4)*b^(3/4)*c^(5/2))
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 = 0.66 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 B}{a \sqrt {c x}}-\frac {2 A c}{3 a \left (c x \right )^{\frac {3}{2}}}+\frac {2 \left (\frac {\left (-A b c +C a c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (-B b +D a \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a}}{c^{2}}\) | \(332\) |
| default | \(\frac {-\frac {2 B}{a \sqrt {c x}}-\frac {2 A c}{3 a \left (c x \right )^{\frac {3}{2}}}+\frac {2 \left (\frac {\left (-A b c +C a c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (-B b +D a \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a}}{c^{2}}\) | \(332\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (c x \right )^{\frac {3}{2}} \left (A b -C a \right ) b \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\frac {a \,c^{2}}{b}}+2 \left (\left (c x \right )^{\frac {3}{2}} \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )}{2}\right ) \left (B b -D a \right ) \sqrt {2}+\frac {4 \left (3 B x +A \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} c b}{3}\right ) c a}{4 \left (c x \right )^{\frac {3}{2}} \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} a^{2} c^{3} b}\) | \(336\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
2/c^2*(-1/a*B/(c*x)^(1/2)-1/3*A*c/a/(c*x)^(3/2)+1/a*(1/8*(-A*b*c+C*a*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*(-B*b+D*a)/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. 3409 vs. \(2 (220) = 440\).
Time = 0.36 (sec) , antiderivative size = 3409, normalized size of antiderivative = 11.56 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\left (c x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(c*x)**(5/2)/(b*x**2+a),x)
Output:
Integral((A + B*x + C*x**2 + D*x**3)/((c*x)**(5/2)*(a + b*x**2)), x)
Time = 0.12 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.28 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {\frac {3 \, {\left (\frac {\sqrt {2} {\left ({\left (D a - B b\right )} \sqrt {a} c - {\left (C a \sqrt {b} - A b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x + \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left ({\left (D a - B b\right )} \sqrt {a} c - {\left (C a \sqrt {b} - A b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x - \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {2 \, \sqrt {2} {\left ({\left (D a - B b\right )} \sqrt {a} c + {\left (C a \sqrt {b} - A b^{\frac {3}{2}}\right )} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c} - \frac {2 \, \sqrt {2} {\left ({\left (D a - B b\right )} \sqrt {a} c + {\left (C a \sqrt {b} - A b^{\frac {3}{2}}\right )} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c}\right )}}{a c} + \frac {8 \, {\left (3 \, B c x + A c\right )}}{\left (c x\right )^{\frac {3}{2}} a c}}{12 \, c} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a),x, algorithm="maxima")
Output:
-1/12*(3*(sqrt(2)*((D*a - B*b)*sqrt(a)*c - (C*a*sqrt(b) - A*b^(3/2))*c)*lo g(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)*((D*a - B*b)*sqrt(a)*c - (C*a*sqrt(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)*((D*a - B*b)*sqrt(a)*c + (C*a*sq rt(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)*sqr t(a)*sqrt(b)*c) - 2*sqrt(2)*((D*a - B*b)*sqrt(a)*c + (C*a*sqrt(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)*s qrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/(sqrt(sqrt(a)*sqrt(b)*c)*sqrt(a)*sqrt(b)* c))/(a*c) + 8*(3*B*c*x + A*c)/((c*x)^(3/2)*a*c))/c
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (220) = 440\).
Time = 0.13 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.61 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, {\left (3 \, B c x + A c\right )}}{3 \, \sqrt {c x} a c^{3} x} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3} c^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3} c^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B b\right )} \log \left (c x + \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{4 \, a^{2} b^{3} c^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B b\right )} \log \left (c x - \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{4 \, a^{2} b^{3} c^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a),x, algorithm="giac")
Output:
-2/3*(3*B*c*x + A*c)/(sqrt(c*x)*a*c^3*x) + 1/2*sqrt(2)*((a*b^3*c^2)^(1/4)* C*a*b^2*c - (a*b^3*c^2)^(1/4)*A*b^3*c + (a*b^3*c^2)^(3/4)*D*a - (a*b^3*c^2 )^(3/4)*B*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^2*b^3*c^4) + 1/2*sqrt(2)*((a*b^3*c^2)^(1/4)*C*a*b^2*c - (a*b^3*c^2)^(1/4)*A*b^3*c + (a*b^3*c^2)^(3/4)*D*a - (a*b^3*c^2)^(3/4)*B*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^2*b^3*c^4) + 1/4*sqrt(2)*((a*b^3*c^2)^(1/4)*C*a*b^2*c - (a*b^3*c^2) ^(1/4)*A*b^3*c - (a*b^3*c^2)^(3/4)*D*a + (a*b^3*c^2)^(3/4)*B*b)*log(c*x + sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sqrt(a*c^2/b))/(a^2*b^3*c^4) - 1/4*sqr t(2)*((a*b^3*c^2)^(1/4)*C*a*b^2*c - (a*b^3*c^2)^(1/4)*A*b^3*c - (a*b^3*c^2 )^(3/4)*D*a + (a*b^3*c^2)^(3/4)*B*b)*log(c*x - sqrt(2)*(a*c^2/b)^(1/4)*sqr t(c*x) + sqrt(a*c^2/b))/(a^2*b^3*c^4)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c\,x\right )}^{5/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(5/2)*(a + b*x^2)),x)
Output:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(5/2)*(a + b*x^2)), x)
Time = 0.16 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.14 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a),x)
Output:
(sqrt(c)*( - 6*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*d*x + 6*sqrt(x) *b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqr t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x + 6*sqrt(x)*b**(3/4)*a**(1/4)*sq rt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1 /4)*sqrt(2)))*a*b*x - 6*sqrt(x)*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*c*x + 6*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sq rt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*d*x - 6*sqrt(x)*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*x - 6*sqrt(x)*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*b*x + 6*sqrt(x)*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*c*x + 3*sqrt(x)*b** (1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d*x - 3*sqrt(x)*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*x - 3*sqrt(x)*b**(1/4) *a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b )*x)*a*d*x + 3*sqrt(x)*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*x + 3*sqrt(x)*b**(3/4)*a**(1/4...