Integrand size = 32, antiderivative size = 405 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {c^2 (5 b B-9 a D) \sqrt {c x}}{2 b^3}-\frac {c (3 A b-7 a C) (c x)^{3/2}}{6 a b^2}+\frac {2 D (c x)^{9/2}}{5 b c^2 \left (a+b x^2\right )}-\frac {(c x)^{5/2} (a (5 b B-9 a D)-5 b (A b-a C) x)}{10 a b^2 \left (a+b x^2\right )}-\frac {c^{5/2} \left (\sqrt {b} (3 A b-7 a C)-\sqrt {a} (5 b B-9 a D)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{13/4}}+\frac {c^{5/2} \left (\sqrt {b} (3 A b-7 a C)-\sqrt {a} (5 b B-9 a D)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{13/4}}-\frac {c^{5/2} \left (\sqrt {b} (3 A b-7 a C)+\sqrt {a} (5 b B-9 a D)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{13/4}} \] Output:
1/2*c^2*(5*B*b-9*D*a)*(c*x)^(1/2)/b^3-1/6*c*(3*A*b-7*C*a)*(c*x)^(3/2)/a/b^ 2+2/5*D*(c*x)^(9/2)/b/c^2/(b*x^2+a)-1/10*(c*x)^(5/2)*(a*(5*B*b-9*D*a)-5*b* (A*b-C*a)*x)/a/b^2/(b*x^2+a)-1/8*c^(5/2)*(b^(1/2)*(3*A*b-7*C*a)-a^(1/2)*(5 *B*b-9*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^(13/4)+1/8*c^(5/2)*(b^(1/2)*(3*A*b-7*C*a)-a^(1/2)*(5*B*b-9*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^ (13/4)-1/8*c^(5/2)*(b^(1/2)*(3*A*b-7*C*a)+a^(1/2)*(5*B*b-9*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^(13/4)
Time = 1.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.64 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {(c x)^{5/2} \left (\frac {4 \sqrt [4]{b} \sqrt {x} \left (-135 a^2 D+a b (75 B+x (35 C-108 D x))+b^2 x \left (-15 A+4 x \left (15 B+5 C x+3 D x^2\right )\right )\right )}{a+b x^2}-\frac {15 \sqrt {2} \left (3 A b^{3/2}-5 \sqrt {a} b B-7 a \sqrt {b} C+9 a^{3/2} D\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {15 \sqrt {2} \left (-3 A b^{3/2}-5 \sqrt {a} b B+7 a \sqrt {b} C+9 a^{3/2} D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}\right )}{120 b^{13/4} x^{5/2}} \] Input:
Integrate[((c*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]
Output:
((c*x)^(5/2)*((4*b^(1/4)*Sqrt[x]*(-135*a^2*D + a*b*(75*B + x*(35*C - 108*D *x)) + b^2*x*(-15*A + 4*x*(15*B + 5*C*x + 3*D*x^2))))/(a + b*x^2) - (15*Sq rt[2]*(3*A*b^(3/2) - 5*Sqrt[a]*b*B - 7*a*Sqrt[b]*C + 9*a^(3/2)*D)*ArcTan[( Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(1/4) + (15*Sqr t[2]*(-3*A*b^(3/2) - 5*Sqrt[a]*b*B + 7*a*Sqrt[b]*C + 9*a^(3/2)*D)*ArcTanh[ (Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4)))/(120*b ^(13/4)*x^(5/2))
Time = 1.24 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2335, 27, 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.
| \(\displaystyle \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {c \int -\frac {(c x)^{3/2} \left (4 a D x^2-(3 A b-7 a C) x+\frac {5 a (b B-a D)}{b}\right )}{2 \left (b x^2+a\right )}dx}{2 a b}-\frac {(c x)^{5/2} \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {(c x)^{3/2} \left (4 a D x^2-(3 A b-7 a C) x+\frac {5 a (b B-a D)}{b}\right )}{b x^2+a}dx}{4 a b}-\frac {(c x)^{5/2} \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {c \int \left (\frac {4 a D (c x)^{3/2}}{b}+\frac {(a (5 b B-9 a D)-b (3 A b-7 a C) x) (c x)^{3/2}}{b \left (b x^2+a\right )}\right )dx}{4 a b}-\frac {(c x)^{5/2} \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c \left (-\frac {a^{3/4} c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ) \left (\sqrt {b} (3 A b-7 a C)-\sqrt {a} (5 b B-9 a D)\right )}{\sqrt {2} b^{9/4}}+\frac {a^{3/4} c^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right ) \left (\sqrt {b} (3 A b-7 a C)-\sqrt {a} (5 b B-9 a D)\right )}{\sqrt {2} b^{9/4}}+\frac {a^{3/4} c^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (3 A b-7 a C)+\sqrt {a} (5 b B-9 a D)\right )}{2 \sqrt {2} b^{9/4}}-\frac {a^{3/4} c^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}+\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {c} x\right ) \left (\sqrt {b} (3 A b-7 a C)+\sqrt {a} (5 b B-9 a D)\right )}{2 \sqrt {2} b^{9/4}}-\frac {2 (c x)^{3/2} (3 A b-7 a C)}{3 b}+\frac {2 a c \sqrt {c x} (5 b B-9 a D)}{b^2}+\frac {8 a D (c x)^{5/2}}{5 b c}\right )}{4 a b}-\frac {(c x)^{5/2} \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
Input:
Int[((c*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]
Output:
-1/2*((c*x)^(5/2)*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(a*b*(a + b*x^2)) + ( c*((2*a*c*(5*b*B - 9*a*D)*Sqrt[c*x])/b^2 - (2*(3*A*b - 7*a*C)*(c*x)^(3/2)) /(3*b) + (8*a*D*(c*x)^(5/2))/(5*b*c) - (a^(3/4)*c^(3/2)*(Sqrt[b]*(3*A*b - 7*a*C) - Sqrt[a]*(5*b*B - 9*a*D))*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/( a^(1/4)*Sqrt[c])])/(Sqrt[2]*b^(9/4)) + (a^(3/4)*c^(3/2)*(Sqrt[b]*(3*A*b - 7*a*C) - Sqrt[a]*(5*b*B - 9*a*D))*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/( a^(1/4)*Sqrt[c])])/(Sqrt[2]*b^(9/4)) + (a^(3/4)*c^(3/2)*(Sqrt[b]*(3*A*b - 7*a*C) + Sqrt[a]*(5*b*B - 9*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]*b^(9/4)) - (a^(3/4)*c^(3/ 2)*(Sqrt[b]*(3*A*b - 7*a*C) + Sqrt[a]*(5*b*B - 9*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]*b^(9 /4))))/(4*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Time = 0.81 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (-\frac {5 \left (b \,x^{2}+a \right ) \left (B b -\frac {9 D a}{5}\right ) \sqrt {2}\, c \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )}{2}\right ) \sqrt {\frac {a \,c^{2}}{b}}}{3}-\frac {4 \left (A b -C a \right ) b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \left (c x \right )^{\frac {3}{2}}}{3}+\left (\frac {20 \left (\frac {4 \left (\frac {1}{5} D x^{2}+\frac {1}{3} C x +B \right ) x^{2} b^{2}}{5}+a \left (-\frac {36}{25} D x^{2}+\frac {4}{15} C x +B \right ) b -\frac {9 D a^{2}}{5}\right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}}{3}+\left (b \,x^{2}+a \right ) \sqrt {2}\, \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )}{2}\right ) c \left (A b -\frac {7 C a}{3}\right )\right ) c \right ) c}{8 \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} b^{3} \left (b \,x^{2}+a \right )}\) | \(402\) |
| derivativedivides | \(\frac {\frac {2 D \left (c x \right )^{\frac {5}{2}} b}{5}+\frac {2 C c \left (c x \right )^{\frac {3}{2}} b}{3}+2 B b \,c^{2} \sqrt {c x}-4 D a \,c^{2} \sqrt {c x}}{b^{3}}+\frac {2 c^{3} \left (\frac {\left (-\frac {1}{4} b^{2} A +\frac {1}{4} C a b \right ) \left (c x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a b B c -\frac {1}{4} D a^{2} c \right ) \sqrt {c x}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\left (-5 a b B c +9 D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,c^{2}}+\frac {\left (3 b^{2} A -7 C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(420\) |
| default | \(\frac {\frac {2 D \left (c x \right )^{\frac {5}{2}} b}{5}+\frac {2 C c \left (c x \right )^{\frac {3}{2}} b}{3}+2 B b \,c^{2} \sqrt {c x}-4 D a \,c^{2} \sqrt {c x}}{b^{3}}+\frac {2 c^{3} \left (\frac {\left (-\frac {1}{4} b^{2} A +\frac {1}{4} C a b \right ) \left (c x \right )^{\frac {3}{2}}+\left (\frac {1}{4} a b B c -\frac {1}{4} D a^{2} c \right ) \sqrt {c x}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\left (-5 a b B c +9 D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,c^{2}}+\frac {\left (3 b^{2} A -7 C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(420\) |
Input:
int((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
3/8/(a*c^2/b)^(1/4)*(-5/3*(b*x^2+a)*(B*b-9/5*D*a)*2^(1/2)*c*(arctan(2^(1/2 )/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1)+arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2 )+1)+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))))*(a*c^2/b)^(1/2)-4/ 3*(A*b-C*a)*b*(a*c^2/b)^(1/4)*(c*x)^(3/2)+(20/3*(4/5*(1/5*D*x^2+1/3*C*x+B) *x^2*b^2+a*(-36/25*D*x^2+4/15*C*x+B)*b-9/5*D*a^2)*(a*c^2/b)^(1/4)*(c*x)^(1 /2)+(b*x^2+a)*2^(1/2)*(arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1)+arcta n(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)+1)+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))))*c*(A*b-7/3*C*a))*c)*c/b^3/(b*x^2+a)
Leaf count of result is larger than twice the leaf count of optimal. 3697 vs. \(2 (314) = 628\).
Time = 0.28 (sec) , antiderivative size = 3697, normalized size of antiderivative = 9.13 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="fricas ")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 107.11 (sec) , antiderivative size = 4189, normalized size of antiderivative = 10.34 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((c*x)**(5/2)*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)
Output:
A*(-21*a**(11/4)*b*c**(5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I *pi/4)/a**(1/4))*gamma(7/4)/(32*a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gam ma(11/4) + 32*a**2*b**(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4)) - 21*I*a **(11/4)*b*c**(5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4) /a**(1/4))*gamma(7/4)/(32*a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/ 4) + 32*a**2*b**(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4)) + 21*a**(11/4) *b*c**(5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4 ))*gamma(7/4)/(32*a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 32* a**2*b**(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4)) + 21*I*a**(11/4)*b*c** (5/2)*x**(11/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gam ma(7/4)/(32*a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 32*a**2*b **(15/4)*x**(15/2)*exp(3*I*pi/4)*gamma(11/4)) - 21*a**(7/4)*b**2*c**(5/2)* x**(15/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(7/4)/ (32*a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 32*a**2*b**(15/4) *x**(15/2)*exp(3*I*pi/4)*gamma(11/4)) - 21*I*a**(7/4)*b**2*c**(5/2)*x**(15 /2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(7/4)/(32* a**3*b**(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 32*a**2*b**(15/4)*x** (15/2)*exp(3*I*pi/4)*gamma(11/4)) + 21*a**(7/4)*b**2*c**(5/2)*x**(15/2)*lo g(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(7/4)/(32*a**3*b **(11/4)*x**(11/2)*exp(3*I*pi/4)*gamma(11/4) + 32*a**2*b**(15/4)*x**(15...
Time = 0.12 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.24 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {15 \, c^{4} {\left (\frac {\sqrt {2} {\left ({\left (7 \, C a b - 3 \, A b^{2}\right )} \sqrt {a} c + {\left (9 \, D a^{2} \sqrt {b} - 5 \, 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 (7 \, C a b - 3 \, A b^{2}\right )} \sqrt {a} c + {\left (9 \, D a^{2} \sqrt {b} - 5 \, 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 (7 \, C a b - 3 \, A b^{2}\right )} \sqrt {a} c - {\left (9 \, D a^{2} \sqrt {b} - 5 \, 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 (7 \, C a b - 3 \, A b^{2}\right )} \sqrt {a} c - {\left (9 \, D a^{2} \sqrt {b} - 5 \, 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^{3}} + \frac {120 \, {\left ({\left (C a b - A b^{2}\right )} \left (c x\right )^{\frac {3}{2}} c^{4} - {\left (D a^{2} - B a b\right )} \sqrt {c x} c^{5}\right )}}{b^{4} c^{2} x^{2} + a b^{3} c^{2}} + \frac {32 \, {\left (3 \, \left (c x\right )^{\frac {5}{2}} D b c + 5 \, \left (c x\right )^{\frac {3}{2}} C b c^{2} - 15 \, {\left (2 \, D a - B b\right )} \sqrt {c x} c^{3}\right )}}{b^{3}}}{240 \, c} \] Input:
integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="maxima ")
Output:
1/240*(15*c^4*(sqrt(2)*((7*C*a*b - 3*A*b^2)*sqrt(a)*c + (9*D*a^2*sqrt(b) - 5*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)*((7*C*a*b - 3*A*b^2)*sqr t(a)*c + (9*D*a^2*sqrt(b) - 5*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*sqr t(2)*((7*C*a*b - 3*A*b^2)*sqrt(a)*c - (9*D*a^2*sqrt(b) - 5*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*s qrt(2)*((7*C*a*b - 3*A*b^2)*sqrt(a)*c - (9*D*a^2*sqrt(b) - 5*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 ^3 + 120*((C*a*b - A*b^2)*(c*x)^(3/2)*c^4 - (D*a^2 - B*a*b)*sqrt(c*x)*c^5) /(b^4*c^2*x^2 + a*b^3*c^2) + 32*(3*(c*x)^(5/2)*D*b*c + 5*(c*x)^(3/2)*C*b*c ^2 - 15*(2*D*a - B*b)*sqrt(c*x)*c^3)/b^3)/c
Time = 0.14 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.52 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*(sqrt(c*x)*C*a*b*c^4*x - sqrt(c*x)*A*b^2*c^4*x - sqrt(c*x)*D*a^2*c^4 + sqrt(c*x)*B*a*b*c^4)/((b*c^2*x^2 + a*c^2)*b^3) + 1/8*sqrt(2)*(9*(a*b^3*c^ 2)^(1/4)*D*a^2*b*c^2 - 5*(a*b^3*c^2)^(1/4)*B*a*b^2*c^2 - 7*(a*b^3*c^2)^(3/ 4)*C*a*c + 3*(a*b^3*c^2)^(3/4)*A*b*c)*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^5) + 1/8*sqrt(2)*(9*(a*b^3*c^ 2)^(1/4)*D*a^2*b*c^2 - 5*(a*b^3*c^2)^(1/4)*B*a*b^2*c^2 - 7*(a*b^3*c^2)^(3/ 4)*C*a*c + 3*(a*b^3*c^2)^(3/4)*A*b*c)*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^5) + 1/16*sqrt(2)*(9*(a*b^3* c^2)^(1/4)*D*a^2*b*c^2 - 5*(a*b^3*c^2)^(1/4)*B*a*b^2*c^2 + 7*(a*b^3*c^2)^( 3/4)*C*a*c - 3*(a*b^3*c^2)^(3/4)*A*b*c)*log(c*x + sqrt(2)*(a*c^2/b)^(1/4)* sqrt(c*x) + sqrt(a*c^2/b))/(a*b^5) - 1/16*sqrt(2)*(9*(a*b^3*c^2)^(1/4)*D*a ^2*b*c^2 - 5*(a*b^3*c^2)^(1/4)*B*a*b^2*c^2 + 7*(a*b^3*c^2)^(3/4)*C*a*c - 3 *(a*b^3*c^2)^(3/4)*A*b*c)*log(c*x - sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sq rt(a*c^2/b))/(a*b^5) + 2/15*(3*sqrt(c*x)*D*b^8*c^2*x^2 + 5*sqrt(c*x)*C*b^8 *c^2*x - 30*sqrt(c*x)*D*a*b^7*c^2 + 15*sqrt(c*x)*B*b^8*c^2)/b^10
Timed out. \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\int \frac {{\left (c\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int(((c*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2)^2,x)
Output:
int(((c*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x^2)^2, x)
Time = 0.16 (sec) , antiderivative size = 1244, normalized size of antiderivative = 3.07 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)
Output:
(sqrt(c)*c**2*( - 90*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2 + 210*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)))*a*b*c - 90*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 **3*x**2 + 210*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*c*x**2 - 270*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**2*d + 150*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**2 - 270*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*d*x**2 + 150*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**3*x**2 + 90*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)))*a *b**2 - 210*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)))*a*b*c + 90*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**3*x**2 - 210*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...