Integrand size = 22, antiderivative size = 120 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=-\frac {2 a}{3 c^2 e (e x)^{3/2}}+\frac {4 a d}{c^3 e^2 \sqrt {e x}}+\frac {\left (b c^2+a d^2\right ) \sqrt {e x}}{c^3 e^3 (c+d x)}+\frac {\left (b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{7/2} \sqrt {d} e^{5/2}} \] Output:
-2/3*a/c^2/e/(e*x)^(3/2)+4*a*d/c^3/e^2/(e*x)^(1/2)+(a*d^2+b*c^2)*(e*x)^(1/ 2)/c^3/e^3/(d*x+c)+(5*a*d^2+b*c^2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1 /2))/c^(7/2)/d^(1/2)/e^(5/2)
Time = 0.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\frac {x \left (\frac {\sqrt {c} \left (3 b c^2 x^2+a \left (-2 c^2+10 c d x+15 d^2 x^2\right )\right )}{c+d x}+\frac {3 \left (b c^2+5 a d^2\right ) x^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{\sqrt {d}}\right )}{3 c^{7/2} (e x)^{5/2}} \] Input:
Integrate[(a + b*x^2)/((e*x)^(5/2)*(c + d*x)^2),x]
Output:
(x*((Sqrt[c]*(3*b*c^2*x^2 + a*(-2*c^2 + 10*c*d*x + 15*d^2*x^2)))/(c + d*x) + (3*(b*c^2 + 5*a*d^2)*x^(3/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/Sqrt[d] ))/(3*c^(7/2)*(e*x)^(5/2))
Time = 0.59 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {518, 1583, 1584, 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.
\(\displaystyle \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 518 |
\(\displaystyle \frac {2 \int \frac {b x^2 e^2+a e^2}{e^2 x^2 (c e+d x e)^2}d\sqrt {e x}}{e}\) |
\(\Big \downarrow \) 1583 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {2 a c^2 d^2 e^4+d^2 \left (b c^2+a d^2\right ) x^2 e^4-2 a c d^3 x e^4}{e^2 x^2 (c e+d x e)}d\sqrt {e x}}{2 c^3 d^2 e^3}+\frac {\sqrt {e x} \left (a d^2+b c^2\right )}{2 c^3 e (c e+d e x)}\right )}{e}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {2 \left (\frac {\int \left (-\frac {4 a e d^3}{x}+\frac {\left (b c^2+5 a d^2\right ) e^2 d^2}{c e+d x e}+\frac {2 a c e d^2}{x^2}\right )d\sqrt {e x}}{2 c^3 d^2 e^3}+\frac {\sqrt {e x} \left (a d^2+b c^2\right )}{2 c^3 e (c e+d e x)}\right )}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {\frac {d^{3/2} e^{3/2} \left (5 a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c}}-\frac {2 a c d^2 e^3}{3 (e x)^{3/2}}+\frac {4 a d^3 e^2}{\sqrt {e x}}}{2 c^3 d^2 e^3}+\frac {\sqrt {e x} \left (a d^2+b c^2\right )}{2 c^3 e (c e+d e x)}\right )}{e}\) |
Input:
Int[(a + b*x^2)/((e*x)^(5/2)*(c + d*x)^2),x]
Output:
(2*(((b*c^2 + a*d^2)*Sqrt[e*x])/(2*c^3*e*(c*e + d*e*x)) + ((-2*a*c*d^2*e^3 )/(3*(e*x)^(3/2)) + (4*a*d^3*e^2)/Sqrt[e*x] + (d^(3/2)*(b*c^2 + 5*a*d^2)*e ^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/Sqrt[c])/(2*c^3*d^2* e^3)))/e
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_Sym bol] :> 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[(-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1)) Int[x^ m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + c*x^4)^p - ((c*d^2 + a*e^2)^p/(e^(m/2)*x^m))*(d + e *(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && IL tQ[q, -1] && ILtQ[m/2, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {2 a \left (-6 d x +c \right )}{3 c^{3} \sqrt {e x}\, x \,e^{2}}+\frac {\frac {2 \left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (5 a \,d^{2}+b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\sqrt {d e c}}}{c^{3} e^{2}}\) | \(96\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {15 \left (d x +c \right ) \left (e x \right )^{\frac {3}{2}} \left (a \,d^{2}+\frac {b \,c^{2}}{5}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2}+\left (\left (-\frac {3 b \,x^{2}}{2}+a \right ) c^{2}-5 a d x c -\frac {15 a \,d^{2} x^{2}}{2}\right ) e \sqrt {d e c}\right )}{3 e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {d e c}\, c^{3} \left (d x +c \right )}\) | \(102\) |
derivativedivides | \(\frac {-\frac {2 a}{3 c^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {4 a d}{e \,c^{3} \sqrt {e x}}+\frac {2 \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (5 a \,d^{2}+b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{3} e}}{e}\) | \(104\) |
default | \(\frac {-\frac {2 a}{3 c^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {4 a d}{e \,c^{3} \sqrt {e x}}+\frac {2 \left (\frac {\left (\frac {a \,d^{2}}{2}+\frac {b \,c^{2}}{2}\right ) \sqrt {e x}}{d e x +c e}+\frac {\left (5 a \,d^{2}+b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 \sqrt {d e c}}\right )}{c^{3} e}}{e}\) | \(104\) |
Input:
int((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-2/3*a*(-6*d*x+c)/c^3/(e*x)^(1/2)/x/e^2+1/c^3*(2*(1/2*a*d^2+1/2*b*c^2)*(e* x)^(1/2)/(d*e*x+c*e)+(5*a*d^2+b*c^2)/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d *e*c)^(1/2)))/e^2
Time = 0.24 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.42 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\left [-\frac {3 \, {\left ({\left (b c^{2} d + 5 \, a d^{3}\right )} x^{3} + {\left (b c^{3} + 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {-c d e} \log \left (\frac {d e x - c e - 2 \, \sqrt {-c d e} \sqrt {e x}}{d x + c}\right ) - 2 \, {\left (10 \, a c^{2} d^{2} x - 2 \, a c^{3} d + 3 \, {\left (b c^{3} d + 5 \, a c d^{3}\right )} x^{2}\right )} \sqrt {e x}}{6 \, {\left (c^{4} d^{2} e^{3} x^{3} + c^{5} d e^{3} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b c^{2} d + 5 \, a d^{3}\right )} x^{3} + {\left (b c^{3} + 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt {c d e} \arctan \left (\frac {\sqrt {c d e} \sqrt {e x}}{d e x}\right ) - {\left (10 \, a c^{2} d^{2} x - 2 \, a c^{3} d + 3 \, {\left (b c^{3} d + 5 \, a c d^{3}\right )} x^{2}\right )} \sqrt {e x}}{3 \, {\left (c^{4} d^{2} e^{3} x^{3} + c^{5} d e^{3} x^{2}\right )}}\right ] \] Input:
integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^2,x, algorithm="fricas")
Output:
[-1/6*(3*((b*c^2*d + 5*a*d^3)*x^3 + (b*c^3 + 5*a*c*d^2)*x^2)*sqrt(-c*d*e)* log((d*e*x - c*e - 2*sqrt(-c*d*e)*sqrt(e*x))/(d*x + c)) - 2*(10*a*c^2*d^2* x - 2*a*c^3*d + 3*(b*c^3*d + 5*a*c*d^3)*x^2)*sqrt(e*x))/(c^4*d^2*e^3*x^3 + c^5*d*e^3*x^2), -1/3*(3*((b*c^2*d + 5*a*d^3)*x^3 + (b*c^3 + 5*a*c*d^2)*x^ 2)*sqrt(c*d*e)*arctan(sqrt(c*d*e)*sqrt(e*x)/(d*e*x)) - (10*a*c^2*d^2*x - 2 *a*c^3*d + 3*(b*c^3*d + 5*a*c*d^3)*x^2)*sqrt(e*x))/(c^4*d^2*e^3*x^3 + c^5* d*e^3*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1452 vs. \(2 (114) = 228\).
Time = 21.16 (sec) , antiderivative size = 1452, normalized size of antiderivative = 12.10 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)/(e*x)**(5/2)/(d*x+c)**2,x)
Output:
a*(15*c**(27/2)*d**2*x**2*atan(sqrt(d)*sqrt(x)/sqrt(c))/(3*c**17*sqrt(d)*e **(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2 )*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) + 60*c**(25/2)*d**3*x**3*atan(sqrt(d)*sqrt(x)/sqrt(c))/(3*c**17*sqrt(d)*e **(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2 )*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) + 90*c**(23/2)*d**4*x**4*atan(sqrt(d)*sqrt(x)/sqrt(c))/(3*c**17*sqrt(d)*e **(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2 )*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) + 60*c**(21/2)*d**5*x**5*atan(sqrt(d)*sqrt(x)/sqrt(c))/(3*c**17*sqrt(d)*e **(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2 )*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) + 15*c**(19/2)*d**6*x**6*atan(sqrt(d)*sqrt(x)/sqrt(c))/(3*c**17*sqrt(d)*e **(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2 )*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) - 2*c**15*sqrt(d)*sqrt(x)/(3*c**17*sqrt(d)*e**(5/2)*x**2 + 12*c**16*d**(3 /2)*e**(5/2)*x**3 + 18*c**15*d**(5/2)*e**(5/2)*x**4 + 12*c**14*d**(7/2)*e* *(5/2)*x**5 + 3*c**13*d**(9/2)*e**(5/2)*x**6) + 4*c**14*d**(3/2)*x**(3/2)/ (3*c**17*sqrt(d)*e**(5/2)*x**2 + 12*c**16*d**(3/2)*e**(5/2)*x**3 + 18*c**1 5*d**(5/2)*e**(5/2)*x**4 + 12*c**14*d**(7/2)*e**(5/2)*x**5 + 3*c**13*d*...
Exception generated. \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^2,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 = 106, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\frac {{\left (b c^{2} + 5 \, a d^{2}\right )} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{\sqrt {c d e} c^{3} e^{2}} + \frac {\sqrt {e x} b c^{2} + \sqrt {e x} a d^{2}}{{\left (d e x + c e\right )} c^{3} e^{2}} + \frac {2 \, {\left (6 \, a d e x - a c e\right )}}{3 \, \sqrt {e x} c^{3} e^{3} x} \] Input:
integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^2,x, algorithm="giac")
Output:
(b*c^2 + 5*a*d^2)*arctan(sqrt(e*x)*d/sqrt(c*d*e))/(sqrt(c*d*e)*c^3*e^2) + (sqrt(e*x)*b*c^2 + sqrt(e*x)*a*d^2)/((d*e*x + c*e)*c^3*e^2) + 2/3*(6*a*d*e *x - a*c*e)/(sqrt(e*x)*c^3*e^3*x)
Time = 6.68 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\frac {\frac {e\,x^2\,\left (b\,c^2+5\,a\,d^2\right )}{c^3}-\frac {2\,a\,e}{3\,c}+\frac {10\,a\,d\,e\,x}{3\,c^2}}{c\,e^2\,{\left (e\,x\right )}^{3/2}+d\,e\,{\left (e\,x\right )}^{5/2}}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e\,x}}{\sqrt {c}\,\sqrt {e}}\right )\,\left (b\,c^2+5\,a\,d^2\right )}{c^{7/2}\,\sqrt {d}\,e^{5/2}} \] Input:
int((a + b*x^2)/((e*x)^(5/2)*(c + d*x)^2),x)
Output:
((e*x^2*(5*a*d^2 + b*c^2))/c^3 - (2*a*e)/(3*c) + (10*a*d*e*x)/(3*c^2))/(c* e^2*(e*x)^(3/2) + d*e*(e*x)^(5/2)) + (atan((d^(1/2)*(e*x)^(1/2))/(c^(1/2)* e^(1/2)))*(5*a*d^2 + b*c^2))/(c^(7/2)*d^(1/2)*e^(5/2))
Time = 0.19 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^2} \, dx=\frac {\sqrt {e}\, \left (15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a c \,d^{2} x +15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a \,d^{3} x^{2}+3 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b \,c^{3} x +3 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b \,c^{2} d \,x^{2}-2 a \,c^{3} d +10 a \,c^{2} d^{2} x +15 a c \,d^{3} x^{2}+3 b \,c^{3} d \,x^{2}\right )}{3 \sqrt {x}\, c^{4} d \,e^{3} x \left (d x +c \right )} \] Input:
int((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^2,x)
Output:
(sqrt(e)*(15*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a *c*d**2*x + 15*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c))) *a*d**3*x**2 + 3*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c) ))*b*c**3*x + 3*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)) )*b*c**2*d*x**2 - 2*a*c**3*d + 10*a*c**2*d**2*x + 15*a*c*d**3*x**2 + 3*b*c **3*d*x**2))/(3*sqrt(x)*c**4*d*e**3*x*(c + d*x))