Integrand size = 28, antiderivative size = 102 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 e \sqrt {e x}}{(b c-a d) \sqrt {a x+b x^2}}-\frac {2 \sqrt {d} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e} \sqrt {a x+b x^2}}{\sqrt {b c-a d} \sqrt {e x}}\right )}{(b c-a d)^{3/2}} \] Output:
-2*e*(e*x)^(1/2)/(-a*d+b*c)/(b*x^2+a*x)^(1/2)-2*d^(1/2)*e^(3/2)*arctan(d^( 1/2)*e^(1/2)*(b*x^2+a*x)^(1/2)/(-a*d+b*c)^(1/2)/(e*x)^(1/2))/(-a*d+b*c)^(3 /2)
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 e \sqrt {e x} \left (\sqrt {b c-a d}+\sqrt {d} \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{(b c-a d)^{3/2} \sqrt {x (a+b x)}} \] Input:
Integrate[(e*x)^(3/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
Output:
(-2*e*Sqrt[e*x]*(Sqrt[b*c - a*d] + Sqrt[d]*Sqrt[a + b*x]*ArcTan[(Sqrt[d]*S qrt[a + b*x])/Sqrt[b*c - a*d]]))/((b*c - a*d)^(3/2)*Sqrt[x*(a + b*x)])
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1261, 61, 73, 218}
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.
\(\displaystyle \int \frac {(e x)^{3/2}}{\left (a x+b x^2\right )^{3/2} (c+d x)} \, dx\) |
\(\Big \downarrow \) 1261 |
\(\displaystyle \frac {(e x)^{3/2} (a+b x)^{3/2} \int \frac {1}{(a+b x)^{3/2} (c+d x)}dx}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {(e x)^{3/2} (a+b x)^{3/2} \left (-\frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)}dx}{b c-a d}-\frac {2}{\sqrt {a+b x} (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(e x)^{3/2} (a+b x)^{3/2} \left (-\frac {2 d \int \frac {1}{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}d\sqrt {a+b x}}{b (b c-a d)}-\frac {2}{\sqrt {a+b x} (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(e x)^{3/2} (a+b x)^{3/2} \left (-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {2}{\sqrt {a+b x} (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[(e*x)^(3/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
Output:
((e*x)^(3/2)*(a + b*x)^(3/2)*(-2/((b*c - a*d)*Sqrt[a + b*x]) - (2*Sqrt[d]* ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2)))/(a*x + b*x^2)^(3/2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {2 e \sqrt {e x}\, \sqrt {x \left (b x +a \right )}\, \left (d \,\operatorname {arctanh}\left (\frac {d \sqrt {\left (b x +a \right ) e}}{\sqrt {e \left (a d -b c \right ) d}}\right ) \sqrt {\left (b x +a \right ) e}-\sqrt {e \left (a d -b c \right ) d}\right )}{x \left (b x +a \right ) \left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\) | \(103\) |
Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*e/x*(e*x)^(1/2)*(x*(b*x+a))^(1/2)*(d*arctanh(d*((b*x+a)*e)^(1/2)/(e*(a* d-b*c)*d)^(1/2))*((b*x+a)*e)^(1/2)-(e*(a*d-b*c)*d)^(1/2))/(b*x+a)/(a*d-b*c )/(e*(a*d-b*c)*d)^(1/2)
Time = 0.14 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.88 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (b e x^{2} + a e x\right )} \sqrt {-\frac {d e}{b c - a d}} \log \left (-\frac {b d e x^{2} - {\left (b c - 2 \, a d\right )} e x + 2 \, \sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {e x} \sqrt {-\frac {d e}{b c - a d}}}{d x^{2} + c x}\right ) + 2 \, \sqrt {b x^{2} + a x} \sqrt {e x} e}{{\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x}, -\frac {2 \, {\left ({\left (b e x^{2} + a e x\right )} \sqrt {\frac {d e}{b c - a d}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {e x} \sqrt {\frac {d e}{b c - a d}}}{b d e x^{2} + a d e x}\right ) + \sqrt {b x^{2} + a x} \sqrt {e x} e\right )}}{{\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x}\right ] \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
[-((b*e*x^2 + a*e*x)*sqrt(-d*e/(b*c - a*d))*log(-(b*d*e*x^2 - (b*c - 2*a*d )*e*x + 2*sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(e*x)*sqrt(-d*e/(b*c - a*d)))/ (d*x^2 + c*x)) + 2*sqrt(b*x^2 + a*x)*sqrt(e*x)*e)/((b^2*c - a*b*d)*x^2 + ( a*b*c - a^2*d)*x), -2*((b*e*x^2 + a*e*x)*sqrt(d*e/(b*c - a*d))*arctan(-sqr t(b*x^2 + a*x)*(b*c - a*d)*sqrt(e*x)*sqrt(d*e/(b*c - a*d))/(b*d*e*x^2 + a* d*e*x)) + sqrt(b*x^2 + a*x)*sqrt(e*x)*e)/((b^2*c - a*b*d)*x^2 + (a*b*c - a ^2*d)*x)]
\[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate((e*x)**(3/2)/(d*x+c)/(b*x**2+a*x)**(3/2),x)
Output:
Integral((e*x)**(3/2)/((x*(a + b*x))**(3/2)*(c + d*x)), x)
\[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x)^(3/2)/((b*x^2 + a*x)^(3/2)*(d*x + c)), x)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (82) = 164\).
Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.00 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-2 \, e^{4} {\left (\frac {d \arctan \left (\frac {\sqrt {b e x + a e} d}{\sqrt {b c d e - a d^{2} e}}\right )}{\sqrt {b c d e - a d^{2} e} {\left (b c e {\left | e \right |} - a d e {\left | e \right |}\right )}} + \frac {1}{{\left (b c e {\left | e \right |} - a d e {\left | e \right |}\right )} \sqrt {b e x + a e}}\right )} + \frac {2 \, {\left (\sqrt {a e} d e^{3} \arctan \left (\frac {\sqrt {a e} d}{\sqrt {b c d e - a d^{2} e}}\right ) + \sqrt {b c d e - a d^{2} e} e^{3}\right )}}{\sqrt {b c d e - a d^{2} e} \sqrt {a e} b c {\left | e \right |} - \sqrt {b c d e - a d^{2} e} \sqrt {a e} a d {\left | e \right |}} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
-2*e^4*(d*arctan(sqrt(b*e*x + a*e)*d/sqrt(b*c*d*e - a*d^2*e))/(sqrt(b*c*d* e - a*d^2*e)*(b*c*e*abs(e) - a*d*e*abs(e))) + 1/((b*c*e*abs(e) - a*d*e*abs (e))*sqrt(b*e*x + a*e))) + 2*(sqrt(a*e)*d*e^3*arctan(sqrt(a*e)*d/sqrt(b*c* d*e - a*d^2*e)) + sqrt(b*c*d*e - a*d^2*e)*e^3)/(sqrt(b*c*d*e - a*d^2*e)*sq rt(a*e)*b*c*abs(e) - sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e)*a*d*abs(e))
Timed out. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int((e*x)^(3/2)/((a*x + b*x^2)^(3/2)*(c + d*x)),x)
Output:
int((e*x)^(3/2)/((a*x + b*x^2)^(3/2)*(c + d*x)), x)
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.85 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {e}\, e \left (-\sqrt {d}\, \sqrt {b x +a}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right )+a d -b c \right )}{\sqrt {b x +a}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \] Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x)
Output:
(2*sqrt(e)*e*( - sqrt(d)*sqrt(a + b*x)*sqrt( - a*d + b*c)*atan((sqrt(a + b *x)*d)/(sqrt(d)*sqrt( - a*d + b*c))) + a*d - b*c))/(sqrt(a + b*x)*(a**2*d* *2 - 2*a*b*c*d + b**2*c**2))