Integrand size = 24, antiderivative size = 284 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 b}+\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} b^{9/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}} \]
-2/3*c^3/a/x^(3/2)+2/5*d^3*x^(5/2)/b+1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^( 1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(9/4)*2^(1/2)-1/2*(-a*d+b*c)^3*arctan(1+b^ (1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(9/4)*2^(1/2)+1/4*(-a*d+b*c)^3*ln (a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/b^(9/4)*2^(1/2 )-1/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a ^(7/4)/b^(9/4)*2^(1/2)+2*d^2*(-a*d+3*b*c)*x^(1/2)/b^2
Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (-5 b^2 c^3-15 a^2 d^3 x^2+3 a b d^2 x^2 \left (15 c+d x^2\right )\right )}{x^{3/2}}+15 \sqrt {2} (b c-a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{30 a^{7/4} b^{9/4}} \]
((4*a^(3/4)*b^(1/4)*(-5*b^2*c^3 - 15*a^2*d^3*x^2 + 3*a*b*d^2*x^2*(15*c + d *x^2)))/x^(3/2) + 15*Sqrt[2]*(b*c - a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*(-(b*c) + a*d)^3*ArcTanh[(Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(30*a^(7/4)*b^(9/4) )
Time = 0.42 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {368, 961, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {\left (d x^2+c\right )^3}{x^2 \left (b x^2+a\right )}d\sqrt {x}\) |
| \(\Big \downarrow \) 961 |
| \(\displaystyle 2 \int \left (\frac {c^3}{a x^2}+\frac {d^3 x^2}{b}+\frac {d^2 (3 b c-a d)}{b^2}+\frac {(a d-b c)^3}{a b^2 \left (b x^2+a\right )}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{2 \sqrt {2} a^{7/4} b^{9/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{2 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} a^{7/4} b^{9/4}}+\frac {d^2 \sqrt {x} (3 b c-a d)}{b^2}-\frac {c^3}{3 a x^{3/2}}+\frac {d^3 x^{5/2}}{5 b}\right )\) |
2*(-1/3*c^3/(a*x^(3/2)) + (d^2*(3*b*c - a*d)*Sqrt[x])/b^2 + (d^3*x^(5/2))/ (5*b) + ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(2*S qrt[2]*a^(7/4)*b^(9/4)) - ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[ x])/a^(1/4)])/(2*Sqrt[2]*a^(7/4)*b^(9/4)) + ((b*c - a*d)^3*Log[Sqrt[a] - S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*a^(7/4)*b^(9/4)) - ((b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] )/(4*Sqrt[2]*a^(7/4)*b^(9/4)))
3.5.46.3.1 Defintions of rubi rules used
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_)), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Time = 2.77 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65
| method | result | size |
| derivativedivides | \(-\frac {2 d^{2} \left (-\frac {b \,x^{\frac {5}{2}} d}{5}+a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}-\frac {2 c^{3}}{3 a \,x^{\frac {3}{2}}}\) | \(186\) |
| default | \(-\frac {2 d^{2} \left (-\frac {b \,x^{\frac {5}{2}} d}{5}+a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}-\frac {2 c^{3}}{3 a \,x^{\frac {3}{2}}}\) | \(186\) |
| risch | \(-\frac {2 \left (-3 a b \,d^{3} x^{4}+15 a^{2} d^{3} x^{2}-45 a b c \,d^{2} x^{2}+5 b^{2} c^{3}\right )}{15 b^{2} x^{\frac {3}{2}} a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}\) | \(198\) |
-2*d^2/b^2*(-1/5*b*x^(5/2)*d+a*d*x^(1/2)-3*b*c*x^(1/2))+1/4/a^2/b^2*(a^3*d ^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(a/b)^(1/4)*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))-2/3*c^3/a/x^(3/2)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1665, normalized size of antiderivative = 5.86 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]
1/30*(15*a*b^2*x^2*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^ 6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^ 3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^7*b^9))^(1 /4)*log(a^2*b^2*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 2 20*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b ^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d ^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^7*b^9))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 15*I*a*b ^2*x^2*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^ 9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^ 6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a ^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^7*b^9))^(1/4)*log(I*a^ 2*b^2*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9 *c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^ 10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^7*b^9))^(1/4) - (b^3*c^ 3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 15*I*a*b^2*x^2*(-( b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792...
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (265) = 530\).
Time = 20.42 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}}{a} & \text {for}\: b = 0 \\- \frac {2 a d^{3} \sqrt {x}}{b^{2}} - \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} + \frac {6 c d^{2} \sqrt {x}}{b} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {2 d^{3} x^{\frac {5}{2}}}{5 b} - \frac {2 c^{3}}{3 a x^{\frac {3}{2}}} - \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*c**3/(7*x**(7/2)) - 2*c**2*d/x**(3/2) + 6*c*d**2*sqrt(x ) + 2*d**3*x**(5/2)/5), Eq(a, 0) & Eq(b, 0)), ((-2*c**3/(7*x**(7/2)) - 2*c **2*d/x**(3/2) + 6*c*d**2*sqrt(x) + 2*d**3*x**(5/2)/5)/b, Eq(a, 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, Eq(b, 0)), (-2*a*d**3*sqrt(x)/b**2 - a*d**3*(-a/b)**(1/4)*log(sqr t(x) - (-a/b)**(1/4))/(2*b**2) + a*d**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b) **(1/4))/(2*b**2) + a*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 + 6*c*d**2*sqrt(x)/b + 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4)) /(2*b) - 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) - 3*c*d **2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 2*d**3*x**(5/2)/(5*b) - 2*c**3/(3*a*x**(3/2)) - 3*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4) )/(2*a) + 3*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a) + 3*c* *2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a + b*c**3*(-a/b)**(1/4)*lo g(sqrt(x) - (-a/b)**(1/4))/(2*a**2) - b*c**3*(-a/b)**(1/4)*log(sqrt(x) + ( -a/b)**(1/4))/(2*a**2) - b*c**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/ a**2, True))
Time = 0.29 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{3 \, a x^{\frac {3}{2}}} + \frac {2 \, {\left (b d^{3} x^{\frac {5}{2}} + 5 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \sqrt {x}\right )}}{5 \, b^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a b^{2}} \]
-2/3*c^3/(a*x^(3/2)) + 2/5*(b*d^3*x^(5/2) + 5*(3*b*c*d^2 - a*d^3)*sqrt(x)) /b^2 - 1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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))) + 2*sqrt(2)*(b^3*c^3 - 3*a*b ^2*c^2*d + 3*a^2*b*c*d^2 - 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)*s qrt(b))) + sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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)) - sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (207) = 414\).
Time = 0.32 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.62 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{3 \, a x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{3}} + \frac {2 \, {\left (b^{4} d^{3} x^{\frac {5}{2}} + 15 \, b^{4} c d^{2} \sqrt {x} - 5 \, a b^{3} d^{3} \sqrt {x}\right )}}{5 \, b^{5}} \]
-2/3*c^3/(a*x^(3/2)) - 1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4 )*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arcta n(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^3) - 1 /2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3 )^(1/4)*a^2*b*c*d^2 - (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^2*b^3) - 1/4*sqrt(2)*((a*b^3)^(1/ 4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - ( a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^ 2*b^3) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x )*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^3) + 2/5*(b^4*d^3*x^(5/2) + 15*b^4*c *d^2*sqrt(x) - 5*a*b^3*d^3*sqrt(x))/b^5
Time = 0.24 (sec) , antiderivative size = 1561, normalized size of antiderivative = 5.50 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]
(2*d^3*x^(5/2))/(5*b) - (2*c^3)/(3*a*x^(3/2)) - x^(1/2)*((2*a*d^3)/b^2 - ( 6*c*d^2)/b) - (atan(((((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4 *b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3* d^3 + 240*a^7*b^11*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8* b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2))/(2*(-a)^(7/4)*b^(9/4))) *(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)) + (((x^(1/2)*(16*a^3*b^15*c^6 + 16 *a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^ 2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a ^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2))/(2 *(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)))/((((x^(1/2)* (16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 4 8*a^7*b^12*c*d^2))/(2*(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3)/((-a)^(7/4)*b^(9 /4)) - (((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7* b^11*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48* a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2))/(2*(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3 )/((-a)^(7/4)*b^(9/4))))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)) - (atan((( ((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^...