Integrand size = 24, antiderivative size = 288 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {2 (b c-a d)^2 \sqrt {x}}{d^3}-\frac {2 b (b c-2 a d) x^{5/2}}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}+\frac {\sqrt [4]{c} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}+\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}} \]
-2/5*b*(-2*a*d+b*c)*x^(5/2)/d^2+2/9*b^2*x^(9/2)/d+1/2*c^(1/4)*(-a*d+b*c)^2 *arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(13/4)*2^(1/2)-1/2*c^(1/4)*(- a*d+b*c)^2*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(13/4)*2^(1/2)+1/4* c^(1/4)*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2)) /d^(13/4)*2^(1/2)-1/4*c^(1/4)*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^ (1/4)*2^(1/2)*x^(1/2))/d^(13/4)*2^(1/2)+2*(-a*d+b*c)^2*x^(1/2)/d^3
Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.65 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {4 \sqrt [4]{d} \sqrt {x} \left (45 a^2 d^2+18 a b d \left (-5 c+d x^2\right )+b^2 \left (45 c^2-9 c d x^2+5 d^2 x^4\right )\right )+45 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-45 \sqrt {2} \sqrt [4]{c} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{90 d^{13/4}} \]
(4*d^(1/4)*Sqrt[x]*(45*a^2*d^2 + 18*a*b*d*(-5*c + d*x^2) + b^2*(45*c^2 - 9 *c*d*x^2 + 5*d^2*x^4)) + 45*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - 45*Sqrt[2]*c^(1/4)*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]) /(90*d^(13/4))
Time = 0.46 (sec) , antiderivative size = 288, 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.
| \(\displaystyle \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 364 |
| \(\displaystyle \int \left (\frac {x^{3/2} \left (a^2 d^2-2 a b c d+b^2 c^2\right )}{d^2 \left (c+d x^2\right )}-\frac {b x^{3/2} (b c-2 a d)}{d^2}+\frac {b^2 x^{7/2}}{d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{c} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{13/4}}+\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}-\frac {\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{13/4}}+\frac {2 \sqrt {x} (b c-a d)^2}{d^3}-\frac {2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac {2 b^2 x^{9/2}}{9 d}\) |
(2*(b*c - a*d)^2*Sqrt[x])/d^3 - (2*b*(b*c - 2*a*d)*x^(5/2))/(5*d^2) + (2*b ^2*x^(9/2))/(9*d) + (c^(1/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqr t[x])/c^(1/4)])/(Sqrt[2]*d^(13/4)) - (c^(1/4)*(b*c - a*d)^2*ArcTan[1 + (Sq rt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(13/4)) + (c^(1/4)*(b*c - a*d) ^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]* d^(13/4)) - (c^(1/4)*(b*c - a*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*S qrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(13/4))
3.5.17.3.1 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 = 2.73 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {2 \left (5 b^{2} d^{2} x^{4}+18 x^{2} a b \,d^{2}-9 x^{2} b^{2} c d +45 a^{2} d^{2}-90 a b c d +45 b^{2} c^{2}\right ) \sqrt {x}}{45 d^{3}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3}}\) | \(190\) |
| derivativedivides | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {2 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-4 a b c d \sqrt {x}+2 b^{2} c^{2} \sqrt {x}}{d^{3}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3}}\) | \(194\) |
| default | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {2 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-4 a b c d \sqrt {x}+2 b^{2} c^{2} \sqrt {x}}{d^{3}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3}}\) | \(194\) |
2/45*(5*b^2*d^2*x^4+18*a*b*d^2*x^2-9*b^2*c*d*x^2+45*a^2*d^2-90*a*b*c*d+45* b^2*c^2)*x^(1/2)/d^3-1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3*(c/d)^(1/4)*2^(1/ 2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)* 2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^( 1/2)/(c/d)^(1/4)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 1131, normalized size of antiderivative = 3.93 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {45 \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} \log \left (d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) + 45 i \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} \log \left (i \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 45 i \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} \log \left (-i \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 45 \, d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} \log \left (-d^{3} \left (-\frac {b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8}}{d^{13}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, b^{2} d^{2} x^{4} + 45 \, b^{2} c^{2} - 90 \, a b c d + 45 \, a^{2} d^{2} - 9 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{2}\right )} \sqrt {x}}{90 \, d^{3}} \]
-1/90*(45*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5 *c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4)*log(d^3*(-(b^8*c^9 - 8*a*b^7*c^8* d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5* b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4 ) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 45*I*d^3*(-(b^8*c^9 - 8*a*b ^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^1 3)^(1/4)*log(I*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^ 3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d ^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2 *d^2)*sqrt(x)) - 45*I*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^ 2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4)*log(-I*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^ 5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c* d^8)/d^13)^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) - 45*d^3*(-(b^ 8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b ^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a ^8*c*d^8)/d^13)^(1/4)*log(-d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*...
Time = 16.13 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.69 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\begin {cases} \tilde {\infty } \left (2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}}{c} & \text {for}\: d = 0 \\\frac {2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}}{d} & \text {for}\: c = 0 \\\frac {2 a^{2} \sqrt {x}}{d} + \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {a^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d} - \frac {4 a b c \sqrt {x}}{d^{2}} - \frac {a b c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d^{2}} + \frac {a b c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d^{2}} + \frac {2 a b c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {4 a b x^{\frac {5}{2}}}{5 d} + \frac {2 b^{2} c^{2} \sqrt {x}}{d^{3}} + \frac {b^{2} c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3}} - \frac {b^{2} c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3}} - \frac {b^{2} c^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3}} - \frac {2 b^{2} c x^{\frac {5}{2}}}{5 d^{2}} + \frac {2 b^{2} x^{\frac {9}{2}}}{9 d} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/2)/9), Eq (c, 0) & Eq(d, 0)), ((2*a**2*x**(5/2)/5 + 4*a*b*x**(9/2)/9 + 2*b**2*x**(13 /2)/13)/c, Eq(d, 0)), ((2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/2 )/9)/d, Eq(c, 0)), (2*a**2*sqrt(x)/d + a**2*(-c/d)**(1/4)*log(sqrt(x) - (- c/d)**(1/4))/(2*d) - a**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d) - a**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/d - 4*a*b*c*sqrt(x)/d**2 - a*b*c*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/d**2 + a*b*c*(-c/d)**( 1/4)*log(sqrt(x) + (-c/d)**(1/4))/d**2 + 2*a*b*c*(-c/d)**(1/4)*atan(sqrt(x )/(-c/d)**(1/4))/d**2 + 4*a*b*x**(5/2)/(5*d) + 2*b**2*c**2*sqrt(x)/d**3 + b**2*c**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*d**3) - b**2*c**2* (-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d**3) - b**2*c**2*(-c/d)**(1 /4)*atan(sqrt(x)/(-c/d)**(1/4))/d**3 - 2*b**2*c*x**(5/2)/(5*d**2) + 2*b**2 *x**(9/2)/(9*d), True))
Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.12 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{4 \, d^{3}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 9 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{3}} \]
-1/4*(2*sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2 )*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqr t(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(-1/ 2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt( d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2* d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d ^(1/4)) - sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^( 1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))*c/d^3 + 2/45*(5*b^2 *d^2*x^(9/2) - 9*(b^2*c*d - 2*a*b*d^2)*x^(5/2) + 45*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x))/d^3
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.34 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{4}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{4}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{4}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{8} x^{\frac {9}{2}} - 9 \, b^{2} c d^{7} x^{\frac {5}{2}} + 18 \, a b d^{8} x^{\frac {5}{2}} + 45 \, b^{2} c^{2} d^{6} \sqrt {x} - 90 \, a b c d^{7} \sqrt {x} + 45 \, a^{2} d^{8} \sqrt {x}\right )}}{45 \, d^{9}} \]
-1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1 /4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1 /4))/d^4 - 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x ))/(c/d)^(1/4))/d^4 - 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4) *a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sq rt(c/d))/d^4 + 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c* d + (c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d ))/d^4 + 2/45*(5*b^2*d^8*x^(9/2) - 9*b^2*c*d^7*x^(5/2) + 18*a*b*d^8*x^(5/2 ) + 45*b^2*c^2*d^6*sqrt(x) - 90*a*b*c*d^7*sqrt(x) + 45*a^2*d^8*sqrt(x))/d^ 9
Time = 4.91 (sec) , antiderivative size = 1175, normalized size of antiderivative = 4.08 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\text {Too large to display} \]
x^(1/2)*((2*a^2)/d + (c*((2*b^2*c)/d^2 - (4*a*b)/d))/d) - x^(5/2)*((2*b^2* c)/(5*d^2) - (4*a*b)/(5*d)) + (2*b^2*x^(9/2))/(9*d) - ((-c)^(1/4)*atan(((( -c)^(1/4)*(a*d - b*c)^2*((8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d ^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 - ((-c)^(1/4)*(a*d - b*c)^2*( 16*b^2*c^4 + 16*a^2*c^2*d^2 - 32*a*b*c^3*d))/(2*d^(13/4)))*1i)/d^(13/4) + ((-c)^(1/4)*(a*d - b*c)^2*((8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3 *d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 + ((-c)^(1/4)*(a*d - b*c)^2 *(16*b^2*c^4 + 16*a^2*c^2*d^2 - 32*a*b*c^3*d))/(2*d^(13/4)))*1i)/d^(13/4)) /(((-c)^(1/4)*(a*d - b*c)^2*((8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c ^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 - ((-c)^(1/4)*(a*d - b*c) ^2*(16*b^2*c^4 + 16*a^2*c^2*d^2 - 32*a*b*c^3*d))/(2*d^(13/4))))/d^(13/4) - ((-c)^(1/4)*(a*d - b*c)^2*((8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^ 3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 + ((-c)^(1/4)*(a*d - b*c)^ 2*(16*b^2*c^4 + 16*a^2*c^2*d^2 - 32*a*b*c^3*d))/(2*d^(13/4))))/d^(13/4)))* (a*d - b*c)^2*1i)/d^(13/4) - ((-c)^(1/4)*atan((((-c)^(1/4)*(a*d - b*c)^2*( (8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 - ((-c)^(1/4)*(a*d - b*c)^2*(16*b^2*c^4 + 16*a^2*c^2*d ^2 - 32*a*b*c^3*d)*1i)/(2*d^(13/4))))/d^(13/4) + ((-c)^(1/4)*(a*d - b*c)^2 *((8*x^(1/2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))/d^3 + ((-c)^(1/4)*(a*d - b*c)^2*(16*b^2*c^4 + 16*a^2*...