Integrand size = 24, antiderivative size = 289 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {2 d (b c-a d)^2 x^{3/2}}{b^4}+\frac {2 d^2 (3 b c-2 a d) x^{7/2}}{7 b^3}+\frac {2 d^3 x^{11/2}}{11 b^2}-\frac {(b c-a d)^3 x^{3/2}}{2 b^4 \left (a+b x^2\right )}-\frac {3 (b c-5 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}-\frac {3 (b c-5 a d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}} \] Output:
2*d*(-a*d+b*c)^2*x^(3/2)/b^4+2/7*d^2*(-2*a*d+3*b*c)*x^(7/2)/b^3+2/11*d^3*x ^(11/2)/b^2-1/2*(-a*d+b*c)^3*x^(3/2)/b^4/(b*x^2+a)-3/8*(-5*a*d+b*c)*(-a*d+ b*c)^2*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(1/4)/b^(19/4)+ 3/8*(-5*a*d+b*c)*(-a*d+b*c)^2*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^ (1/2)/a^(1/4)/b^(19/4)-3/8*(-5*a*d+b*c)*(-a*d+b*c)^2*arctanh(2^(1/2)*a^(1/ 4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(1/4)/b^(19/4)
Time = 0.52 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.89 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 b^{3/4} x^{3/2} \left (385 a^3 d^3+11 a^2 b d^2 \left (-77 c+20 d x^2\right )+a b^2 d \left (539 c^2-484 c d x^2-60 d^2 x^4\right )+b^3 \left (-77 c^3+308 c^2 d x^2+132 c d^2 x^4+28 d^3 x^6\right )\right )}{a+b x^2}+\frac {231 \sqrt {2} (b c-a d)^2 (-b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {231 \sqrt {2} (b c-a d)^2 (-b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}}{616 b^{19/4}} \] Input:
Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]
Output:
((4*b^(3/4)*x^(3/2)*(385*a^3*d^3 + 11*a^2*b*d^2*(-77*c + 20*d*x^2) + a*b^2 *d*(539*c^2 - 484*c*d*x^2 - 60*d^2*x^4) + b^3*(-77*c^3 + 308*c^2*d*x^2 + 1 32*c*d^2*x^4 + 28*d^3*x^6)))/(a + b*x^2) + (231*Sqrt[2]*(b*c - a*d)^2*(-(b *c) + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] )])/a^(1/4) + (231*Sqrt[2]*(b*c - a*d)^2*(-(b*c) + 5*a*d)*ArcTanh[(Sqrt[2] *a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4))/(616*b^(19/4))
Time = 0.55 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {368, 967, 27, 1040, 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.
| \(\displaystyle \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^3 \left (d x^2+c\right )^3}{\left (b x^2+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle 2 \left (\frac {\int \frac {3 x \left (d x^2+c\right )^2 \left (5 d x^2+c\right )}{b x^2+a}d\sqrt {x}}{4 b}-\frac {x^{3/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3 \int \frac {x \left (d x^2+c\right )^2 \left (5 d x^2+c\right )}{b x^2+a}d\sqrt {x}}{4 b}-\frac {x^{3/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle 2 \left (\frac {3 \int \left (\frac {5 d^3 x^5}{b}+\frac {d^2 (11 b c-5 a d) x^3}{b^2}+\frac {d \left (7 b^2 c^2-11 a b d c+5 a^2 d^2\right ) x}{b^3}+\frac {\left (b^3 c^3-7 a b^2 d c^2+11 a^2 b d^2 c-5 a^3 d^3\right ) x}{b^3 \left (b x^2+a\right )}\right )d\sqrt {x}}{4 b}-\frac {x^{3/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{3 b^3}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-5 a d) (b c-a d)^2}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-5 a d) (b c-a d)^2}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-5 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-5 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {d^2 x^{7/2} (11 b c-5 a d)}{7 b^2}+\frac {5 d^3 x^{11/2}}{11 b}\right )}{4 b}-\frac {x^{3/2} \left (c+d x^2\right )^3}{4 b \left (a+b x^2\right )}\right )\) |
Input:
Int[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]
Output:
2*(-1/4*(x^(3/2)*(c + d*x^2)^3)/(b*(a + b*x^2)) + (3*((d*(7*b^2*c^2 - 11*a *b*c*d + 5*a^2*d^2)*x^(3/2))/(3*b^3) + (d^2*(11*b*c - 5*a*d)*x^(7/2))/(7*b ^2) + (5*d^3*x^(11/2))/(11*b) - ((b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (S qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - 5 *a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt [2]*a^(1/4)*b^(15/4)) + ((b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2] *a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b* c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S qrt[b]*x])/(4*Sqrt[2]*a^(1/4)*b^(15/4))))/(4*b))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*n*(p + 1))), x] - Simp[e^n/(b*n*(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*( q - 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBino mialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ (g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c , d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Time = 0.56 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(\frac {2 x^{\frac {3}{2}} d \left (7 b^{2} d^{2} x^{4}-22 x^{2} a b \,d^{2}+33 x^{2} b^{2} c d +77 a^{2} d^{2}-154 a b c d +77 b^{2} c^{2}\right )}{77 b^{4}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15 a d}{4}-\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\) | \(230\) |
| derivativedivides | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15}{4} a^{3} d^{3}-\frac {33}{4} a^{2} b c \,d^{2}+\frac {21}{4} a \,b^{2} c^{2} d -\frac {3}{4} b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\) | \(266\) |
| default | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15}{4} a^{3} d^{3}-\frac {33}{4} a^{2} b c \,d^{2}+\frac {21}{4} a \,b^{2} c^{2} d -\frac {3}{4} b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\) | \(266\) |
Input:
int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2/77*x^(3/2)*d*(7*b^2*d^2*x^4-22*a*b*d^2*x^2+33*b^2*c*d*x^2+77*a^2*d^2-154 *a*b*c*d+77*b^2*c^2)/b^4-1/b^4*(2*a^2*d^2-4*a*b*c*d+2*b^2*c^2)*((-1/4*a*d+ 1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(15/4*a*d-3/4*b*c)/b/(a/b)^(1/4)*2^(1/2)*(l n((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/ 2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/ (a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 2104, normalized size of antiderivative = 7.28 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/616*(231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^ 10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7 *c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c ^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d ^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(27*a*b^14*(-(b^12*c^12 - 28*a*b^1 1*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^ 8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5* d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d ^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5* c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d ^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) + 231*(-I*b^5*x^2 - I*a*b^4)* (-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9 *d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d ^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19)) ^(1/4)*log(27*I*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10 *d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^ 5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 ...
Timed out. \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**(5/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.17 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {3 \, {\left (b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 11 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, b^{4}} + \frac {2 \, {\left (7 \, b^{2} d^{3} x^{\frac {11}{2}} + 11 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{77 \, b^{4}} \] Input:
integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^(3/2)/(b^5*x^2 + a*b^4) + 3/16*(b^3*c^3 - 7*a*b^2*c^2*d + 11*a^2*b*c*d^2 - 5*a^3*d^3)*(2* sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/s qrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(- 1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqr t(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/ 4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2) *a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/b^4 + 2 /77*(7*b^2*d^3*x^(11/2) + 11*(3*b^2*c*d^2 - 2*a*b*d^3)*x^(7/2) + 77*(b^2*c ^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^(3/2))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (223) = 446\).
Time = 0.14 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.91 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="giac")
Output:
-1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^ 3*d^3*x^(3/2))/((b*x^2 + a)*b^4) + 3/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7* (a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4) *a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4) )/(a*b^7) + 3/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2 *d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*s qrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^7) - 3/16*sqrt( 2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4) *a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^( 3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3) *log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 2/77*(7*b^20* d^3*x^(11/2) + 33*b^20*c*d^2*x^(7/2) - 22*a*b^19*d^3*x^(7/2) + 77*b^20*c^2 *d*x^(3/2) - 154*a*b^19*c*d^2*x^(3/2) + 77*a^2*b^18*d^3*x^(3/2))/b^22
Time = 0.27 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.36 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{3/2}\,\left (\frac {2\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )}{3\,b}-\frac {2\,a^2\,d^3}{3\,b^4}\right )-x^{7/2}\,\left (\frac {4\,a\,d^3}{7\,b^3}-\frac {6\,c\,d^2}{7\,b^2}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b^2}+\frac {x^{3/2}\,\left (\frac {a^3\,d^3}{2}-\frac {3\,a^2\,b\,c\,d^2}{2}+\frac {3\,a\,b^2\,c^2\,d}{2}-\frac {b^3\,c^3}{2}\right )}{b^5\,x^2+a\,b^4}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,\left (25\,a^7\,d^6-110\,a^6\,b\,c\,d^5+191\,a^5\,b^2\,c^2\,d^4-164\,a^4\,b^3\,c^3\,d^3+71\,a^3\,b^4\,c^4\,d^2-14\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (125\,a^{10}\,d^9-825\,a^9\,b\,c\,d^8+2340\,a^8\,b^2\,c^2\,d^7-3716\,a^7\,b^3\,c^3\,d^6+3606\,a^6\,b^4\,c^4\,d^5-2190\,a^5\,b^5\,c^5\,d^4+820\,a^4\,b^6\,c^6\,d^3-180\,a^3\,b^7\,c^7\,d^2+21\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )}{4\,{\left (-a\right )}^{1/4}\,b^{19/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,\left (25\,a^7\,d^6-110\,a^6\,b\,c\,d^5+191\,a^5\,b^2\,c^2\,d^4-164\,a^4\,b^3\,c^3\,d^3+71\,a^3\,b^4\,c^4\,d^2-14\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (125\,a^{10}\,d^9-825\,a^9\,b\,c\,d^8+2340\,a^8\,b^2\,c^2\,d^7-3716\,a^7\,b^3\,c^3\,d^6+3606\,a^6\,b^4\,c^4\,d^5-2190\,a^5\,b^5\,c^5\,d^4+820\,a^4\,b^6\,c^6\,d^3-180\,a^3\,b^7\,c^7\,d^2+21\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,3{}\mathrm {i}}{4\,{\left (-a\right )}^{1/4}\,b^{19/4}} \] Input:
int((x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x)
Output:
x^(3/2)*((2*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/(3*b) - (2* a^2*d^3)/(3*b^4)) - x^(7/2)*((4*a*d^3)/(7*b^3) - (6*c*d^2)/(7*b^2)) + (2*d ^3*x^(11/2))/(11*b^2) + (x^(3/2)*((a^3*d^3)/2 - (b^3*c^3)/2 + (3*a*b^2*c^2 *d)/2 - (3*a^2*b*c*d^2)/2))/(a*b^4 + b^5*x^2) - (3*atan((b^(1/4)*x^(1/2)*( a*d - b*c)^2*(5*a*d - b*c)*(25*a^7*d^6 + a*b^6*c^6 - 14*a^2*b^5*c^5*d + 71 *a^3*b^4*c^4*d^2 - 164*a^4*b^3*c^3*d^3 + 191*a^5*b^2*c^2*d^4 - 110*a^6*b*c *d^5))/((-a)^(1/4)*(125*a^10*d^9 - a*b^9*c^9 + 21*a^2*b^8*c^8*d - 180*a^3* b^7*c^7*d^2 + 820*a^4*b^6*c^6*d^3 - 2190*a^5*b^5*c^5*d^4 + 3606*a^6*b^4*c^ 4*d^5 - 3716*a^7*b^3*c^3*d^6 + 2340*a^8*b^2*c^2*d^7 - 825*a^9*b*c*d^8)))*( a*d - b*c)^2*(5*a*d - b*c))/(4*(-a)^(1/4)*b^(19/4)) - (atan((b^(1/4)*x^(1/ 2)*(a*d - b*c)^2*(5*a*d - b*c)*(25*a^7*d^6 + a*b^6*c^6 - 14*a^2*b^5*c^5*d + 71*a^3*b^4*c^4*d^2 - 164*a^4*b^3*c^3*d^3 + 191*a^5*b^2*c^2*d^4 - 110*a^6 *b*c*d^5)*1i)/((-a)^(1/4)*(125*a^10*d^9 - a*b^9*c^9 + 21*a^2*b^8*c^8*d - 1 80*a^3*b^7*c^7*d^2 + 820*a^4*b^6*c^6*d^3 - 2190*a^5*b^5*c^5*d^4 + 3606*a^6 *b^4*c^4*d^5 - 3716*a^7*b^3*c^3*d^6 + 2340*a^8*b^2*c^2*d^7 - 825*a^9*b*c*d ^8)))*(a*d - b*c)^2*(5*a*d - b*c)*3i)/(4*(-a)^(1/4)*b^(19/4))
Time = 0.22 (sec) , antiderivative size = 1405, normalized size of antiderivative = 4.86 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x)
Output:
(2310*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**4*d**3 - 5082*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**3*b*c*d**2 + 2310*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* *3*b*d**3*x**2 + 3234*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*c**2*d - 5082*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**2*b**2*c*d**2*x**2 - 462*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**3*c**3 + 3234*b**(1/4)*a**(3/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr t(2)))*a*b**3*c**2*d*x**2 - 462*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**4*c** 3*x**2 - 2310*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**4*d**3 + 5082*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**3*b*c*d**2 - 2310*b**(1/4)*a**(3/4)*sqrt(2)*at an((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt (2)))*a**3*b*d**3*x**2 - 3234*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*...