Integrand size = 24, antiderivative size = 306 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac {2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}-\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}} \]
2/3*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^(3/2)/b^3+2/7*d^2*(-a*d+3*b*c)*x^(7/ 2)/b^2+2/11*d^3*x^(11/2)/b-1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/ 2)/a^(1/4))/a^(1/4)/b^(15/4)*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^(1/4)/b^(15/4)*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^(1/4)/b^(15/4)*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^(1/4)/ b^(15/4)*2^(1/2)
Time = 0.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d x^{3/2} \left (77 a^2 d^2-33 a b d \left (7 c+d x^2\right )+3 b^2 \left (77 c^2+33 c d x^2+7 d^2 x^4\right )\right )}{231 b^3}+\frac {(-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{15/4}} \]
(2*d*x^(3/2)*(77*a^2*d^2 - 33*a*b*d*(7*c + d*x^2) + 3*b^2*(77*c^2 + 33*c*d *x^2 + 7*d^2*x^4)))/(231*b^3) + ((-(b*c) + a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b ]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((-( b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b] *x)])/(Sqrt[2]*a^(1/4)*b^(15/4))
Time = 0.47 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {364, 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.
| \(\displaystyle \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 364 |
| \(\displaystyle \int \left (\frac {d \sqrt {x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {\sqrt {x} \left (-a^3 d^3+3 a^2 b c d^2-3 a b^2 c^2 d+b^3 c^3\right )}{b^3 \left (a+b x^2\right )}+\frac {d^2 x^{5/2} (3 b c-a d)}{b^2}+\frac {d^3 x^{9/2}}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac {2 d^3 x^{11/2}}{11 b}\) |
(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11/2))/(11*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^ (15/4)) + ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S qrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*Log[Sqrt[a] + Sqr t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4))
3.5.43.3.1 Defintions of rubi rules used
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Time = 2.76 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {2 \left (21 b^{2} d^{2} x^{4}-33 x^{2} a b \,d^{2}+99 x^{2} b^{2} c d +77 a^{2} d^{2}-231 a b c d +231 b^{2} c^{2}\right ) d \,x^{\frac {3}{2}}}{231 b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(206\) |
| derivativedivides | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(208\) |
| default | \(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(208\) |
2/231*(21*b^2*d^2*x^4-33*a*b*d^2*x^2+99*b^2*c*d*x^2+77*a^2*d^2-231*a*b*c*d +231*b^2*c^2)*d*x^(3/2)/b^3-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c ^3)/b^4/(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))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1992, normalized size of antiderivative = 6.51 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\text {Too large to display} \]
-1/462*(231*b^3*(-(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*b^15))^(1/4)* log(a*b^11*(-(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*b^15))^(3/4) - (b^ 9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4* b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^ 7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) - 231*I*b^3*(-(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*b^15))^(1/4)*log(I*a*b^11*(-(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^1 1 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7 *d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 +...
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (291) = 582\).
Time = 17.09 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{\sqrt {x}} + 2 c^{2} d x^{\frac {3}{2}} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7} + \frac {2 d^{3} x^{\frac {11}{2}}}{11}}{b} & \text {for}\: a = 0 \\\frac {\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}}{a} & \text {for}\: b = 0 \\\frac {2 a^{2} d^{3} x^{\frac {3}{2}}}{3 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {a^{2} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} - \frac {2 a c d^{2} x^{\frac {3}{2}}}{b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {3 a c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {2 a d^{3} x^{\frac {7}{2}}}{7 b^{2}} + \frac {2 c^{2} d x^{\frac {3}{2}}}{b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} + \frac {3 c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c d^{2} x^{\frac {7}{2}}}{7 b} + \frac {2 d^{3} x^{\frac {11}{2}}}{11 b} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*c**3/sqrt(x) + 2*c**2*d*x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11), Eq(a, 0) & Eq(b, 0)), ((-2*c**3/sqrt(x) + 2*c**2*d *x**(3/2) + 6*c*d**2*x**(7/2)/7 + 2*d**3*x**(11/2)/11)/b, Eq(a, 0)), ((2*c **3*x**(3/2)/3 + 6*c**2*d*x**(7/2)/7 + 6*c*d**2*x**(11/2)/11 + 2*d**3*x**( 15/2)/15)/a, Eq(b, 0)), (2*a**2*d**3*x**(3/2)/(3*b**3) + a**2*d**3*(-a/b)* *(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**3) - a**2*d**3*(-a/b)**(3/4)*log (sqrt(x) + (-a/b)**(1/4))/(2*b**3) + a**2*d**3*(-a/b)**(3/4)*atan(sqrt(x)/ (-a/b)**(1/4))/b**3 - 2*a*c*d**2*x**(3/2)/b**2 - 3*a*c*d**2*(-a/b)**(3/4)* log(sqrt(x) - (-a/b)**(1/4))/(2*b**2) + 3*a*c*d**2*(-a/b)**(3/4)*log(sqrt( x) + (-a/b)**(1/4))/(2*b**2) - 3*a*c*d**2*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b )**(1/4))/b**2 - 2*a*d**3*x**(7/2)/(7*b**2) + 2*c**2*d*x**(3/2)/b + 3*c**2 *d*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - 3*c**2*d*(-a/b)**(3/ 4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) + 3*c**2*d*(-a/b)**(3/4)*atan(sqrt(x )/(-a/b)**(1/4))/b + 6*c*d**2*x**(7/2)/(7*b) + 2*d**3*x**(11/2)/(11*b) - c **3*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + c**3*(-a/b)**(3/4)* log(sqrt(x) + (-a/b)**(1/4))/(2*a) - c**3*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b )**(1/4))/a, True))
Time = 0.29 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{3}} + \frac {2 \, {\left (21 \, b^{2} d^{3} x^{\frac {11}{2}} + 33 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{231 \, b^{3}} \]
1/4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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))/sqrt(sqrt(a)*sqr t(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sq rt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sq rt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq rt(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^3 + 2/231*(21*b^2*d^ 3*x^(11/2) + 33*(3*b^2*c*d^2 - a*b*d^3)*x^(7/2) + 77*(3*b^2*c^2*d - 3*a*b* c*d^2 + a^2*d^3)*x^(3/2))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (227) = 454\).
Time = 0.30 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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 b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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 b^{6}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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 b^{6}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{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 b^{6}} + \frac {2 \, {\left (21 \, b^{10} d^{3} x^{\frac {11}{2}} + 99 \, b^{10} c d^{2} x^{\frac {7}{2}} - 33 \, a b^{9} d^{3} x^{\frac {7}{2}} + 231 \, b^{10} c^{2} d x^{\frac {3}{2}} - 231 \, a b^{9} c d^{2} x^{\frac {3}{2}} + 77 \, a^{2} b^{8} d^{3} x^{\frac {3}{2}}\right )}}{231 \, b^{11}} \]
1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^ 3)^(3/4)*a^2*b*c*d^2 - (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^6) + 1/2*sqrt(2)*((a*b^3)^(3/4) *b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (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^6) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4 )*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(s qrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^6) + 1/4*sqrt(2)*((a*b^3) ^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b) )/(a*b^6) + 2/231*(21*b^10*d^3*x^(11/2) + 99*b^10*c*d^2*x^(7/2) - 33*a*b^9 *d^3*x^(7/2) + 231*b^10*c^2*d*x^(3/2) - 231*a*b^9*c*d^2*x^(3/2) + 77*a^2*b ^8*d^3*x^(3/2))/b^11
Time = 0.18 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {x} \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{3/2}\,\left (\frac {2\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{3\,b}\right )-x^{7/2}\,\left (\frac {2\,a\,d^3}{7\,b^2}-\frac {6\,c\,d^2}{7\,b}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^7\,d^6-6\,a^6\,b\,c\,d^5+15\,a^5\,b^2\,c^2\,d^4-20\,a^4\,b^3\,c^3\,d^3+15\,a^3\,b^4\,c^4\,d^2-6\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (a^{10}\,d^9-9\,a^9\,b\,c\,d^8+36\,a^8\,b^2\,c^2\,d^7-84\,a^7\,b^3\,c^3\,d^6+126\,a^6\,b^4\,c^4\,d^5-126\,a^5\,b^5\,c^5\,d^4+84\,a^4\,b^6\,c^6\,d^3-36\,a^3\,b^7\,c^7\,d^2+9\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,b^{15/4}} \]
x^(3/2)*((2*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(3*b)) - x^(7/2)* ((2*a*d^3)/(7*b^2) - (6*c*d^2)/(7*b)) + (2*d^3*x^(11/2))/(11*b) - (atan((b ^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a ^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5)) /((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9 + 9*a^2*b^8*c^8*d - 36*a^3*b^7*c^7*d^2 + 84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 126*a^6*b^4*c^4*d^5 - 84*a^7* b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a^9*b*c*d^8)))*(a*d - b*c)^3)/((-a)^( 1/4)*b^(15/4)) - (atan((b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^7*d^6 + a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2* c^2*d^4 - 6*a^6*b*c*d^5)*1i)/((-a)^(1/4)*(a^10*d^9 - a*b^9*c^9 + 9*a^2*b^8 *c^8*d - 36*a^3*b^7*c^7*d^2 + 84*a^4*b^6*c^6*d^3 - 126*a^5*b^5*c^5*d^4 + 1 26*a^6*b^4*c^4*d^5 - 84*a^7*b^3*c^3*d^6 + 36*a^8*b^2*c^2*d^7 - 9*a^9*b*c*d ^8)))*(a*d - b*c)^3*1i)/((-a)^(1/4)*b^(15/4))