Integrand size = 24, antiderivative size = 249 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=-\frac {4 b c \left (5 b c^2+3 a d^2\right ) e \sqrt {e x}}{d^6}+\frac {4 b \left (3 b c^2+a d^2\right ) (e x)^{3/2}}{3 d^5}-\frac {6 b^2 c (e x)^{5/2}}{5 d^4 e}+\frac {2 b^2 (e x)^{7/2}}{7 d^3 e^2}+\frac {c \left (b c^2+a d^2\right )^2 e \sqrt {e x}}{2 d^6 (c+d x)^2}-\frac {\left (b c^2+a d^2\right ) \left (21 b c^2+5 a d^2\right ) e \sqrt {e x}}{4 d^6 (c+d x)}+\frac {\left (99 b^2 c^4+70 a b c^2 d^2+3 a^2 d^4\right ) e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 \sqrt {c} d^{13/2}} \] Output:
-4*b*c*(3*a*d^2+5*b*c^2)*e*(e*x)^(1/2)/d^6+4/3*b*(a*d^2+3*b*c^2)*(e*x)^(3/ 2)/d^5-6/5*b^2*c*(e*x)^(5/2)/d^4/e+2/7*b^2*(e*x)^(7/2)/d^3/e^2+1/2*c*(a*d^ 2+b*c^2)^2*e*(e*x)^(1/2)/d^6/(d*x+c)^2-1/4*(a*d^2+b*c^2)*(5*a*d^2+21*b*c^2 )*e*(e*x)^(1/2)/d^6/(d*x+c)+1/4*(3*a^2*d^4+70*a*b*c^2*d^2+99*b^2*c^4)*e^(3 /2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(1/2)/d^(13/2)
Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.81 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\frac {(e x)^{3/2} \left (\frac {\sqrt {d} \sqrt {x} \left (-105 a^2 d^4 (3 c+5 d x)-70 a b d^2 \left (105 c^3+175 c^2 d x+56 c d^2 x^2-8 d^3 x^3\right )-3 b^2 \left (3465 c^5+5775 c^4 d x+1848 c^3 d^2 x^2-264 c^2 d^3 x^3+88 c d^4 x^4-40 d^5 x^5\right )\right )}{(c+d x)^2}+\frac {105 \left (99 b^2 c^4+70 a b c^2 d^2+3 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{420 d^{13/2} x^{3/2}} \] Input:
Integrate[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x)^3,x]
Output:
((e*x)^(3/2)*((Sqrt[d]*Sqrt[x]*(-105*a^2*d^4*(3*c + 5*d*x) - 70*a*b*d^2*(1 05*c^3 + 175*c^2*d*x + 56*c*d^2*x^2 - 8*d^3*x^3) - 3*b^2*(3465*c^5 + 5775* c^4*d*x + 1848*c^3*d^2*x^2 - 264*c^2*d^3*x^3 + 88*c*d^4*x^4 - 40*d^5*x^5)) )/(c + d*x)^2 + (105*(99*b^2*c^4 + 70*a*b*c^2*d^2 + 3*a^2*d^4)*ArcTan[(Sqr t[d]*Sqrt[x])/Sqrt[c]])/Sqrt[c]))/(420*d^(13/2)*x^(3/2))
Time = 1.54 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {518, 1581, 2345, 2341, 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.
| \(\displaystyle \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 518 |
| \(\displaystyle \frac {2 \int \frac {e^2 x^2 \left (b x^2 e^2+a e^2\right )^2}{(c e+d x e)^3}d\sqrt {e x}}{e^2}\) |
| \(\Big \downarrow \) 1581 |
| \(\displaystyle \frac {2 \left (\frac {c e^5 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 d^6 (c e+d e x)^2}-\frac {\int \frac {-4 b^2 d^5 x^5 e^5+4 b^2 c d^4 x^4 e^5-4 b d^3 \left (b c^2+2 a d^2\right ) x^3 e^5+c \left (b c^2+a d^2\right )^2 e^5+4 b c d^2 \left (b c^2+2 a d^2\right ) x^2 e^5-4 d \left (b c^2+a d^2\right )^2 x e^5}{(c e+d x e)^2}d\sqrt {e x}}{4 d^6}\right )}{e^2}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {2 \left (\frac {c e^5 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 d^6 (c e+d e x)^2}-\frac {\frac {e^4 \sqrt {e x} \left (a d^2+b c^2\right ) \left (5 a d^2+21 b c^2\right )}{2 (c e+d e x)}-\frac {\int \frac {8 b^2 c d^4 x^4 e^5-16 b^2 c^2 d^3 x^3 e^5+8 b c d^2 \left (3 b c^2+2 a d^2\right ) x^2 e^5+c \left (b c^2+a d^2\right ) \left (19 b c^2+3 a d^2\right ) e^5-32 b c^2 d \left (b c^2+a d^2\right ) x e^5}{c e+d x e}d\sqrt {e x}}{2 c e}}{4 d^6}\right )}{e^2}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \frac {2 \left (\frac {c e^5 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 d^6 (c e+d e x)^2}-\frac {\frac {e^4 \sqrt {e x} \left (a d^2+b c^2\right ) \left (5 a d^2+21 b c^2\right )}{2 (c e+d e x)}-\frac {\int \left (8 b^2 c d^3 x^3 e^4-24 b^2 c^2 d^2 x^2 e^4-16 b c^2 \left (5 b c^2+3 a d^2\right ) e^4+16 b c d \left (3 b c^2+a d^2\right ) x e^4+\frac {99 b^2 c^5 e^5+3 a^2 c d^4 e^5+70 a b c^3 d^2 e^5}{c e+d x e}\right )d\sqrt {e x}}{2 c e}}{4 d^6}\right )}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {c e^5 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 d^6 (c e+d e x)^2}-\frac {\frac {e^4 \sqrt {e x} \left (a d^2+b c^2\right ) \left (5 a d^2+21 b c^2\right )}{2 (c e+d e x)}-\frac {\frac {\sqrt {c} e^{9/2} \left (3 a^2 d^4+70 a b c^2 d^2+99 b^2 c^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {d}}-16 b c^2 e^4 \sqrt {e x} \left (3 a d^2+5 b c^2\right )+\frac {16}{3} b c d e^3 (e x)^{3/2} \left (a d^2+3 b c^2\right )-\frac {24}{5} b^2 c^2 d^2 e^2 (e x)^{5/2}+\frac {8}{7} b^2 c d^3 e (e x)^{7/2}}{2 c e}}{4 d^6}\right )}{e^2}\) |
Input:
Int[((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x)^3,x]
Output:
(2*((c*(b*c^2 + a*d^2)^2*e^5*Sqrt[e*x])/(4*d^6*(c*e + d*e*x)^2) - (((b*c^2 + a*d^2)*(21*b*c^2 + 5*a*d^2)*e^4*Sqrt[e*x])/(2*(c*e + d*e*x)) - (-16*b*c ^2*(5*b*c^2 + 3*a*d^2)*e^4*Sqrt[e*x] + (16*b*c*d*(3*b*c^2 + a*d^2)*e^3*(e* x)^(3/2))/3 - (24*b^2*c^2*d^2*e^2*(e*x)^(5/2))/5 + (8*b^2*c*d^3*e*(e*x)^(7 /2))/7 + (Sqrt[c]*(99*b^2*c^4 + 70*a*b*c^2*d^2 + 3*a^2*d^4)*e^(9/2)*ArcTan [(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/Sqrt[d])/(2*c*e))/(4*d^6)))/e^2
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^(2*m + 1)*(e*c + d*x^2)^ n*(a*e^2 + b*x^4)^p, x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Sy mbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^ (2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)*(q + 1)) Int[(d + e*x ^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*e^(2*p + m/2)*(q + 1)*x ^m*(a + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*(d + e*(2*q + 3)*x^2))] , x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ [m/2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.76
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\sqrt {d e c}\, \left (\left (-\frac {8}{21} x^{5} d^{5}+\frac {88}{105} x^{4} c \,d^{4}-\frac {88}{35} c^{2} d^{3} x^{3}+\frac {88}{5} c^{3} d^{2} x^{2}+55 c^{4} d x +33 c^{5}\right ) b^{2}+\frac {70 d^{2} \left (-\frac {8}{105} d^{3} x^{3}+\frac {8}{15} c \,d^{2} x^{2}+\frac {5}{3} c^{2} d x +c^{3}\right ) a b}{3}+a^{2} d^{4} \left (\frac {5 d x}{3}+c \right )\right ) \sqrt {e x}-\left (d x +c \right )^{2} e \left (a^{2} d^{4}+\frac {70}{3} b \,c^{2} d^{2} a +33 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )\right ) e}{4 \sqrt {d e c}\, d^{6} \left (d x +c \right )^{2}}\) | \(188\) |
| risch | \(-\frac {2 b \left (-15 b \,d^{3} x^{3}+63 b c \,d^{2} x^{2}-70 a x \,d^{3}-210 b \,c^{2} d x +630 a \,d^{2} c +1050 b \,c^{3}\right ) x \,e^{2}}{105 d^{6} \sqrt {e x}}+\frac {\left (\frac {2 \left (-\frac {5}{8} a^{2} d^{5}-\frac {13}{4} d^{3} a \,c^{2} b -\frac {21}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+2 \left (-\frac {11}{4} a b \,d^{2} e \,c^{3}-\frac {19}{8} b^{2} e \,c^{5}-\frac {3}{8} c e \,a^{2} d^{4}\right ) \sqrt {e x}}{\left (d e x +c e \right )^{2}}+\frac {\left (3 a^{2} d^{4}+70 b \,c^{2} d^{2} a +99 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{4 \sqrt {d e c}}\right ) e^{2}}{d^{6}}\) | \(206\) |
| derivativedivides | \(\frac {-\frac {2 b \left (-\frac {b \,d^{3} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {3 b c \,d^{2} e \left (e x \right )^{\frac {5}{2}}}{5}-\frac {2 a \,d^{3} e^{2} \left (e x \right )^{\frac {3}{2}}}{3}-2 b \,c^{2} d \,e^{2} \left (e x \right )^{\frac {3}{2}}+6 a c \,d^{2} e^{3} \sqrt {e x}+10 b \,c^{3} e^{3} \sqrt {e x}\right )}{d^{6}}+\frac {2 e^{4} \left (\frac {\left (-\frac {5}{8} a^{2} d^{5}-\frac {13}{4} d^{3} a \,c^{2} b -\frac {21}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (-\frac {11}{4} a b \,d^{2} e \,c^{3}-\frac {19}{8} b^{2} e \,c^{5}-\frac {3}{8} c e \,a^{2} d^{4}\right ) \sqrt {e x}}{\left (d e x +c e \right )^{2}}+\frac {\left (3 a^{2} d^{4}+70 b \,c^{2} d^{2} a +99 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{d^{6}}}{e^{2}}\) | \(236\) |
| default | \(\frac {-\frac {2 b \left (-\frac {b \,d^{3} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {3 b c \,d^{2} e \left (e x \right )^{\frac {5}{2}}}{5}-\frac {2 a \,d^{3} e^{2} \left (e x \right )^{\frac {3}{2}}}{3}-2 b \,c^{2} d \,e^{2} \left (e x \right )^{\frac {3}{2}}+6 a c \,d^{2} e^{3} \sqrt {e x}+10 b \,c^{3} e^{3} \sqrt {e x}\right )}{d^{6}}+\frac {2 e^{4} \left (\frac {\left (-\frac {5}{8} a^{2} d^{5}-\frac {13}{4} d^{3} a \,c^{2} b -\frac {21}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\left (-\frac {11}{4} a b \,d^{2} e \,c^{3}-\frac {19}{8} b^{2} e \,c^{5}-\frac {3}{8} c e \,a^{2} d^{4}\right ) \sqrt {e x}}{\left (d e x +c e \right )^{2}}+\frac {\left (3 a^{2} d^{4}+70 b \,c^{2} d^{2} a +99 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{d^{6}}}{e^{2}}\) | \(236\) |
Input:
int((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-3/4*((d*e*c)^(1/2)*((-8/21*x^5*d^5+88/105*x^4*c*d^4-88/35*c^2*d^3*x^3+88/ 5*c^3*d^2*x^2+55*c^4*d*x+33*c^5)*b^2+70/3*d^2*(-8/105*d^3*x^3+8/15*c*d^2*x ^2+5/3*c^2*d*x+c^3)*a*b+a^2*d^4*(5/3*d*x+c))*(e*x)^(1/2)-(d*x+c)^2*e*(a^2* d^4+70/3*b*c^2*d^2*a+33*b^2*c^4)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))/(d*e *c)^(1/2)*e/d^6/(d*x+c)^2
Time = 0.25 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.55 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\left [\frac {105 \, {\left ({\left (99 \, b^{2} c^{4} d^{2} + 70 \, a b c^{2} d^{4} + 3 \, a^{2} d^{6}\right )} e x^{2} + 2 \, {\left (99 \, b^{2} c^{5} d + 70 \, a b c^{3} d^{3} + 3 \, a^{2} c d^{5}\right )} e x + {\left (99 \, b^{2} c^{6} + 70 \, a b c^{4} d^{2} + 3 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {-\frac {e}{c d}} \log \left (\frac {d e x + 2 \, \sqrt {e x} c d \sqrt {-\frac {e}{c d}} - c e}{d x + c}\right ) + 2 \, {\left (120 \, b^{2} d^{5} e x^{5} - 264 \, b^{2} c d^{4} e x^{4} + 8 \, {\left (99 \, b^{2} c^{2} d^{3} + 70 \, a b d^{5}\right )} e x^{3} - 56 \, {\left (99 \, b^{2} c^{3} d^{2} + 70 \, a b c d^{4}\right )} e x^{2} - 175 \, {\left (99 \, b^{2} c^{4} d + 70 \, a b c^{2} d^{3} + 3 \, a^{2} d^{5}\right )} e x - 105 \, {\left (99 \, b^{2} c^{5} + 70 \, a b c^{3} d^{2} + 3 \, a^{2} c d^{4}\right )} e\right )} \sqrt {e x}}{840 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}}, \frac {105 \, {\left ({\left (99 \, b^{2} c^{4} d^{2} + 70 \, a b c^{2} d^{4} + 3 \, a^{2} d^{6}\right )} e x^{2} + 2 \, {\left (99 \, b^{2} c^{5} d + 70 \, a b c^{3} d^{3} + 3 \, a^{2} c d^{5}\right )} e x + {\left (99 \, b^{2} c^{6} + 70 \, a b c^{4} d^{2} + 3 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {\frac {e}{c d}} \arctan \left (\frac {\sqrt {e x} d \sqrt {\frac {e}{c d}}}{e}\right ) + {\left (120 \, b^{2} d^{5} e x^{5} - 264 \, b^{2} c d^{4} e x^{4} + 8 \, {\left (99 \, b^{2} c^{2} d^{3} + 70 \, a b d^{5}\right )} e x^{3} - 56 \, {\left (99 \, b^{2} c^{3} d^{2} + 70 \, a b c d^{4}\right )} e x^{2} - 175 \, {\left (99 \, b^{2} c^{4} d + 70 \, a b c^{2} d^{3} + 3 \, a^{2} d^{5}\right )} e x - 105 \, {\left (99 \, b^{2} c^{5} + 70 \, a b c^{3} d^{2} + 3 \, a^{2} c d^{4}\right )} e\right )} \sqrt {e x}}{420 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}}\right ] \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^3,x, algorithm="fricas")
Output:
[1/840*(105*((99*b^2*c^4*d^2 + 70*a*b*c^2*d^4 + 3*a^2*d^6)*e*x^2 + 2*(99*b ^2*c^5*d + 70*a*b*c^3*d^3 + 3*a^2*c*d^5)*e*x + (99*b^2*c^6 + 70*a*b*c^4*d^ 2 + 3*a^2*c^2*d^4)*e)*sqrt(-e/(c*d))*log((d*e*x + 2*sqrt(e*x)*c*d*sqrt(-e/ (c*d)) - c*e)/(d*x + c)) + 2*(120*b^2*d^5*e*x^5 - 264*b^2*c*d^4*e*x^4 + 8* (99*b^2*c^2*d^3 + 70*a*b*d^5)*e*x^3 - 56*(99*b^2*c^3*d^2 + 70*a*b*c*d^4)*e *x^2 - 175*(99*b^2*c^4*d + 70*a*b*c^2*d^3 + 3*a^2*d^5)*e*x - 105*(99*b^2*c ^5 + 70*a*b*c^3*d^2 + 3*a^2*c*d^4)*e)*sqrt(e*x))/(d^8*x^2 + 2*c*d^7*x + c^ 2*d^6), 1/420*(105*((99*b^2*c^4*d^2 + 70*a*b*c^2*d^4 + 3*a^2*d^6)*e*x^2 + 2*(99*b^2*c^5*d + 70*a*b*c^3*d^3 + 3*a^2*c*d^5)*e*x + (99*b^2*c^6 + 70*a*b *c^4*d^2 + 3*a^2*c^2*d^4)*e)*sqrt(e/(c*d))*arctan(sqrt(e*x)*d*sqrt(e/(c*d) )/e) + (120*b^2*d^5*e*x^5 - 264*b^2*c*d^4*e*x^4 + 8*(99*b^2*c^2*d^3 + 70*a *b*d^5)*e*x^3 - 56*(99*b^2*c^3*d^2 + 70*a*b*c*d^4)*e*x^2 - 175*(99*b^2*c^4 *d + 70*a*b*c^2*d^3 + 3*a^2*d^5)*e*x - 105*(99*b^2*c^5 + 70*a*b*c^3*d^2 + 3*a^2*c*d^4)*e)*sqrt(e*x))/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)]
\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \] Input:
integrate((e*x)**(3/2)*(b*x**2+a)**2/(d*x+c)**3,x)
Output:
Integral((e*x)**(3/2)*(a + b*x**2)**2/(c + d*x)**3, x)
Exception generated. \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.20 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\frac {1}{420} \, e {\left (\frac {105 \, {\left (99 \, b^{2} c^{4} e + 70 \, a b c^{2} d^{2} e + 3 \, a^{2} d^{4} e\right )} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{\sqrt {c d e} d^{6}} - \frac {105 \, {\left (21 \, \sqrt {e x} b^{2} c^{4} d e^{2} x + 26 \, \sqrt {e x} a b c^{2} d^{3} e^{2} x + 5 \, \sqrt {e x} a^{2} d^{5} e^{2} x + 19 \, \sqrt {e x} b^{2} c^{5} e^{2} + 22 \, \sqrt {e x} a b c^{3} d^{2} e^{2} + 3 \, \sqrt {e x} a^{2} c d^{4} e^{2}\right )}}{{\left (d e x + c e\right )}^{2} d^{6}} + \frac {8 \, {\left (15 \, \sqrt {e x} b^{2} d^{18} e^{21} x^{3} - 63 \, \sqrt {e x} b^{2} c d^{17} e^{21} x^{2} + 210 \, \sqrt {e x} b^{2} c^{2} d^{16} e^{21} x + 70 \, \sqrt {e x} a b d^{18} e^{21} x - 1050 \, \sqrt {e x} b^{2} c^{3} d^{15} e^{21} - 630 \, \sqrt {e x} a b c d^{17} e^{21}\right )}}{d^{21} e^{21}}\right )} \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^3,x, algorithm="giac")
Output:
1/420*e*(105*(99*b^2*c^4*e + 70*a*b*c^2*d^2*e + 3*a^2*d^4*e)*arctan(sqrt(e *x)*d/sqrt(c*d*e))/(sqrt(c*d*e)*d^6) - 105*(21*sqrt(e*x)*b^2*c^4*d*e^2*x + 26*sqrt(e*x)*a*b*c^2*d^3*e^2*x + 5*sqrt(e*x)*a^2*d^5*e^2*x + 19*sqrt(e*x) *b^2*c^5*e^2 + 22*sqrt(e*x)*a*b*c^3*d^2*e^2 + 3*sqrt(e*x)*a^2*c*d^4*e^2)/( (d*e*x + c*e)^2*d^6) + 8*(15*sqrt(e*x)*b^2*d^18*e^21*x^3 - 63*sqrt(e*x)*b^ 2*c*d^17*e^21*x^2 + 210*sqrt(e*x)*b^2*c^2*d^16*e^21*x + 70*sqrt(e*x)*a*b*d ^18*e^21*x - 1050*sqrt(e*x)*b^2*c^3*d^15*e^21 - 630*sqrt(e*x)*a*b*c*d^17*e ^21)/(d^21*e^21))
Time = 0.13 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.39 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\sqrt {e\,x}\,\left (\frac {16\,b^2\,c^3\,e}{d^6}-\frac {3\,c\,e\,\left (\frac {12\,b^2\,c^2}{d^5}+\frac {4\,a\,b}{d^3}\right )}{d}\right )-\frac {\sqrt {e\,x}\,\left (\frac {3\,a^2\,c\,d^4\,e^3}{4}+\frac {11\,a\,b\,c^3\,d^2\,e^3}{2}+\frac {19\,b^2\,c^5\,e^3}{4}\right )+{\left (e\,x\right )}^{3/2}\,\left (\frac {5\,a^2\,d^5\,e^2}{4}+\frac {13\,a\,b\,c^2\,d^3\,e^2}{2}+\frac {21\,b^2\,c^4\,d\,e^2}{4}\right )}{c^2\,d^6\,e^2+2\,c\,d^7\,e^2\,x+d^8\,e^2\,x^2}+{\left (e\,x\right )}^{3/2}\,\left (\frac {4\,b^2\,c^2}{d^5}+\frac {4\,a\,b}{3\,d^3}\right )+\frac {2\,b^2\,{\left (e\,x\right )}^{7/2}}{7\,d^3\,e^2}+\frac {e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,e^{3/2}\,\sqrt {e\,x}\,\left (3\,a^2\,d^4+70\,a\,b\,c^2\,d^2+99\,b^2\,c^4\right )}{\sqrt {c}\,\left (3\,a^2\,d^4\,e^2+70\,a\,b\,c^2\,d^2\,e^2+99\,b^2\,c^4\,e^2\right )}\right )\,\left (3\,a^2\,d^4+70\,a\,b\,c^2\,d^2+99\,b^2\,c^4\right )}{4\,\sqrt {c}\,d^{13/2}}-\frac {6\,b^2\,c\,{\left (e\,x\right )}^{5/2}}{5\,d^4\,e} \] Input:
int(((e*x)^(3/2)*(a + b*x^2)^2)/(c + d*x)^3,x)
Output:
(e*x)^(1/2)*((16*b^2*c^3*e)/d^6 - (3*c*e*((12*b^2*c^2)/d^5 + (4*a*b)/d^3)) /d) - ((e*x)^(1/2)*((19*b^2*c^5*e^3)/4 + (3*a^2*c*d^4*e^3)/4 + (11*a*b*c^3 *d^2*e^3)/2) + (e*x)^(3/2)*((5*a^2*d^5*e^2)/4 + (21*b^2*c^4*d*e^2)/4 + (13 *a*b*c^2*d^3*e^2)/2))/(c^2*d^6*e^2 + d^8*e^2*x^2 + 2*c*d^7*e^2*x) + (e*x)^ (3/2)*((4*b^2*c^2)/d^5 + (4*a*b)/(3*d^3)) + (2*b^2*(e*x)^(7/2))/(7*d^3*e^2 ) + (e^(3/2)*atan((d^(1/2)*e^(3/2)*(e*x)^(1/2)*(3*a^2*d^4 + 99*b^2*c^4 + 7 0*a*b*c^2*d^2))/(c^(1/2)*(3*a^2*d^4*e^2 + 99*b^2*c^4*e^2 + 70*a*b*c^2*d^2* e^2)))*(3*a^2*d^4 + 99*b^2*c^4 + 70*a*b*c^2*d^2))/(4*c^(1/2)*d^(13/2)) - ( 6*b^2*c*(e*x)^(5/2))/(5*d^4*e)
Time = 0.17 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.79 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^3} \, dx=\frac {\sqrt {e}\, e \left (315 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c^{2} d^{4}+630 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{5} x +315 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{6} x^{2}+7350 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{4} d^{2}+14700 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d^{3} x +7350 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{4} x^{2}+10395 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{6}+20790 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{5} d x +10395 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4} d^{2} x^{2}-315 \sqrt {x}\, a^{2} c^{2} d^{5}-525 \sqrt {x}\, a^{2} c \,d^{6} x -7350 \sqrt {x}\, a b \,c^{4} d^{3}-12250 \sqrt {x}\, a b \,c^{3} d^{4} x -3920 \sqrt {x}\, a b \,c^{2} d^{5} x^{2}+560 \sqrt {x}\, a b c \,d^{6} x^{3}-10395 \sqrt {x}\, b^{2} c^{6} d -17325 \sqrt {x}\, b^{2} c^{5} d^{2} x -5544 \sqrt {x}\, b^{2} c^{4} d^{3} x^{2}+792 \sqrt {x}\, b^{2} c^{3} d^{4} x^{3}-264 \sqrt {x}\, b^{2} c^{2} d^{5} x^{4}+120 \sqrt {x}\, b^{2} c \,d^{6} x^{5}\right )}{420 c \,d^{7} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^3,x)
Output:
(sqrt(e)*e*(315*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a**2*c **2*d**4 + 630*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a**2*c* d**5*x + 315*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a**2*d**6 *x**2 + 7350*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**4* d**2 + 14700*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**3* d**3*x + 7350*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**2 *d**4*x**2 + 10395*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b** 2*c**6 + 20790*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c* *5*d*x + 10395*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c* *4*d**2*x**2 - 315*sqrt(x)*a**2*c**2*d**5 - 525*sqrt(x)*a**2*c*d**6*x - 73 50*sqrt(x)*a*b*c**4*d**3 - 12250*sqrt(x)*a*b*c**3*d**4*x - 3920*sqrt(x)*a* b*c**2*d**5*x**2 + 560*sqrt(x)*a*b*c*d**6*x**3 - 10395*sqrt(x)*b**2*c**6*d - 17325*sqrt(x)*b**2*c**5*d**2*x - 5544*sqrt(x)*b**2*c**4*d**3*x**2 + 792 *sqrt(x)*b**2*c**3*d**4*x**3 - 264*sqrt(x)*b**2*c**2*d**5*x**4 + 120*sqrt( x)*b**2*c*d**6*x**5))/(420*c*d**7*(c**2 + 2*c*d*x + d**2*x**2))