Integrand size = 24, antiderivative size = 263 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {2 c^3}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b^2}-\frac {(b c-a d)^3 \sqrt {x}}{2 a^2 b^2 \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}} \] Output:
-2/3*c^3/a^2/x^(3/2)+2*d^3*x^(1/2)/b^2-1/2*(-a*d+b*c)^3*x^(1/2)/a^2/b^2/(b *x^2+a)+1/8*(-a*d+b*c)^2*(5*a*d+7*b*c)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^ (1/4))*2^(1/2)/a^(11/4)/b^(9/4)-1/8*(-a*d+b*c)^2*(5*a*d+7*b*c)*arctan(1+2^ (1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)/b^(9/4)-1/8*(-a*d+b*c)^2*( 5*a*d+7*b*c)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))* 2^(1/2)/a^(11/4)/b^(9/4)
Time = 0.49 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (-7 b^3 c^3 x^2+15 a^3 d^3 x^2+3 a^2 b d^2 x^2 \left (-3 c+4 d x^2\right )+a b^2 c^2 \left (-4 c+9 d x^2\right )\right )}{x^{3/2} \left (a+b x^2\right )}+3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{11/4} b^{9/4}} \] Input:
Integrate[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]
Output:
((4*a^(3/4)*b^(1/4)*(-7*b^3*c^3*x^2 + 15*a^3*d^3*x^2 + 3*a^2*b*d^2*x^2*(-3 *c + 4*d*x^2) + a*b^2*c^2*(-4*c + 9*d*x^2)))/(x^(3/2)*(a + b*x^2)) + 3*Sqr t[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[x])] - 3*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTanh [(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(24*a^(11/4)*b^ (9/4))
Time = 0.55 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {368, 968, 25, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {\left (d x^2+c\right )^3}{x^2 \left (b x^2+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 968 |
| \(\displaystyle 2 \left (\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 a b x^{3/2} \left (a+b x^2\right )}-\frac {\int -\frac {\left (d x^2+c\right ) \left (c (7 b c-3 a d)-d (b c-5 a d) x^2\right )}{x^2 \left (b x^2+a\right )}d\sqrt {x}}{4 a b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {\left (d x^2+c\right ) \left (c (7 b c-3 a d)-d (b c-5 a d) x^2\right )}{x^2 \left (b x^2+a\right )}d\sqrt {x}}{4 a b}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 a b x^{3/2} \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle 2 \left (\frac {\int \left (-\frac {(3 a d-7 b c) c^2}{a x^2}-\frac {d^2 (b c-5 a d)}{b}-\frac {(a d-b c)^2 (7 b c+5 a d)}{a b \left (b x^2+a\right )}\right )d\sqrt {x}}{4 a b}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 a b x^{3/2} \left (a+b x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (5 a d+7 b c)}{2 \sqrt {2} a^{7/4} b^{5/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (5 a d+7 b c)}{2 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {c^2 (7 b c-3 a d)}{3 a x^{3/2}}-\frac {d^2 \sqrt {x} (b c-5 a d)}{b}}{4 a b}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 a b x^{3/2} \left (a+b x^2\right )}\right )\) |
Input:
Int[(c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x]
Output:
2*(((b*c - a*d)*(c + d*x^2)^2)/(4*a*b*x^(3/2)*(a + b*x^2)) + (-1/3*(c^2*(7 *b*c - 3*a*d))/(a*x^(3/2)) - (d^2*(b*c - 5*a*d)*Sqrt[x])/b + ((b*c - a*d)^ 2*(7*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2 ]*a^(7/4)*b^(5/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^ (1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*a^(7/4)*b^(5/4)) + ((b*c - a*d)^2*(7*b *c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4 *Sqrt[2]*a^(7/4)*b^(5/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] + S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*a^(7/4)*b^(5/4)))/ (4*a*b))
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[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) *((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1)) Int [(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c *b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[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 = 0.58 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(\frac {2 a^{2} d^{3} x^{2}-\frac {2 b^{2} c^{3}}{3}}{b^{2} x^{\frac {3}{2}} a^{2}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a d +7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\) | \(201\) |
| derivativedivides | \(\frac {2 d^{3} \sqrt {x}}{b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\) | \(225\) |
| default | \(\frac {2 d^{3} \sqrt {x}}{b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\) | \(225\) |
Input:
int((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2/3*(3*a^2*d^3*x^2-b^2*c^3)/b^2/x^(3/2)/a^2-1/a^2/b^2*(2*a^2*d^2-4*a*b*c*d +2*b^2*c^2)*((-1/4*a*d+1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(5*a*d+7*b*c)*(a/b) ^(1/4)/a*2^(1/2)*(ln((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 = 0.11 (sec) , antiderivative size = 1778, normalized size of antiderivative = 6.76 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/24*(3*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11 *d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^ 4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(a^3*b^2*(-(2401* b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c ^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6 *d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^ 9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9 ))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x) ) + 3*(I*a^2*b^3*x^4 + I*a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^1 1*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d ^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(I*a^3*b^2*(-(24 01*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^ 9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6* c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3 *d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11* b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*s...
Leaf count of result is larger than twice the leaf count of optimal. 2008 vs. \(2 (245) = 490\).
Time = 106.58 (sec) , antiderivative size = 2008, normalized size of antiderivative = 7.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x**2+c)**3/x**(5/2)/(b*x**2+a)**2,x)
Output:
Piecewise((zoo*(-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/ x**(3/2) + 2*d**3*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9)/a**2, Eq(b, 0 )), ((-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x))/b**2, Eq(a, 0)), (15*a**4*d**3*x**(3/2)*(-a/b)**(1/4)*log (sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 15*a**4*d**3*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a** 4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 30*a**4*d**3*x**(3/2)*(-a/b)**( 1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x** (7/2)) + 60*a**4*d**3*x**2/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 9*a**3*b*c*d**2*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24 *a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 9*a**3*b*c*d**2*x**(3/2)*(- a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3* b**3*x**(7/2)) + 18*a**3*b*c*d**2*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/ b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 36*a**3*b*c*d **2*x**2/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 15*a**3*b*d**3* x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 15*a**3*b*d**3*x**(7/2)*(-a/b)**(1/4)*log(sqrt (x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 30* a**3*b*d**3*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4...
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (201) = 402\).
Time = 0.13 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.58 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{2}}{6 \, {\left (a^{2} b^{3} x^{\frac {7}{2}} + a^{3} b^{2} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{2} b^{2}} \] Input:
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
2*d^3*sqrt(x)/b^2 - 1/6*(4*a*b^2*c^3 + (7*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2* b*c*d^2 - 3*a^3*d^3)*x^2)/(a^2*b^3*x^(7/2) + a^3*b^2*x^(3/2)) - 1/16*(2*sq rt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*arctan(1/2*s qrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)) )/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*log(sqr t(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sq rt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*log(-sqrt(2) *a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a^2*b^ 2)
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (201) = 402\).
Time = 0.14 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.90 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {2 \, c^{3}}{3 \, a^{2} x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{3}} - \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{2} b^{2}} \] Input:
integrate((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
2*d^3*sqrt(x)/b^2 - 2/3*c^3/(a^2*x^(3/2)) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b ^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a* b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/( a/b)^(1/4))/(a^3*b^3) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^( 1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)* arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^ 3) - 1/16*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x )*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) + 1/16*sqrt(2)*(7*(a*b^3)^(1/4)*b ^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a* b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3 *b^3) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt( x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a^2*b^2)
Time = 0.31 (sec) , antiderivative size = 1759, normalized size of antiderivative = 6.69 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
int((c + d*x^2)^3/(x^(5/2)*(a + b*x^2)^2),x)
Output:
((x^2*(3*a^3*d^3 - 7*b^3*c^3 + 9*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(6*a^2) - ( 2*b^2*c^3)/(3*a))/(b^3*x^(7/2) + a*b^2*x^(3/2)) + (2*d^3*x^(1/2))/b^2 - (a tan((((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2 592*a^10*b^11*c^2*d^4) - ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(8*(- a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11/4)*b^(9/ 4)) + ((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5* d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) + ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^ 3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(8*( -a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11/4)*b^(9 /4)))/(((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5 *d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) - ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c ^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(8* (-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c))/(8*(-a)^(11/4)*b^(9/4 )) - ((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2 592*a^10*b^11*c^2*d^4) + ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*...
Time = 0.21 (sec) , antiderivative size = 1424, normalized size of antiderivative = 5.41 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^3/x^(5/2)/(b*x^2+a)^2,x)
Output:
(30*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**4*d**3*x - 18*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**3*b*c*d**2*x + 30*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**3*b*d**3*x**3 - 54*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**2*b**2*c**2*d*x - 18*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**2*b**2*c*d**2*x**3 + 42*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 **3*c**3*x - 54*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**3*c**2*d*x* *3 + 42*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)))*b**4*c**3*x**3 - 30*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**4*d**3*x + 18*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**3*b*c*d**2*x - 30*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**(...