Integrand size = 28, antiderivative size = 83 \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\frac {d e \sqrt {a x^2+b x^3}}{b \sqrt {e x}}+\frac {(2 b c-a d) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a x^2+b x^3}}\right )}{b^{3/2}} \] Output:
d*e*(b*x^3+a*x^2)^(1/2)/b/(e*x)^(1/2)+(-a*d+2*b*c)*e^(1/2)*arctanh(b^(1/2) *(e*x)^(3/2)/e^(3/2)/(b*x^3+a*x^2)^(1/2))/b^(3/2)
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\frac {\sqrt {x} \sqrt {e x} \left (\sqrt {b} d \sqrt {x} (a+b x)+2 (2 b c-a d) \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{b^{3/2} \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(Sqrt[e*x]*(c + d*x))/Sqrt[a*x^2 + b*x^3],x]
Output:
(Sqrt[x]*Sqrt[e*x]*(Sqrt[b]*d*Sqrt[x]*(a + b*x) + 2*(2*b*c - a*d)*Sqrt[a + b*x]*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(b^(3/2)*Sqr t[x^2*(a + b*x)])
Time = 0.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1945, 1937, 1935, 219}
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.
\(\displaystyle \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx\) |
\(\Big \downarrow \) 1945 |
\(\displaystyle \frac {(2 b c-a d) \int \frac {\sqrt {e x}}{\sqrt {b x^3+a x^2}}dx}{2 b}+\frac {d e \sqrt {a x^2+b x^3}}{b \sqrt {e x}}\) |
\(\Big \downarrow \) 1937 |
\(\displaystyle \frac {\sqrt {e x} (2 b c-a d) \int \frac {\sqrt {x}}{\sqrt {b x^3+a x^2}}dx}{2 b \sqrt {x}}+\frac {d e \sqrt {a x^2+b x^3}}{b \sqrt {e x}}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {\sqrt {e x} (2 b c-a d) \int \frac {1}{1-\frac {b x^3}{b x^3+a x^2}}d\frac {x^{3/2}}{\sqrt {b x^3+a x^2}}}{b \sqrt {x}}+\frac {d e \sqrt {a x^2+b x^3}}{b \sqrt {e x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {e x} \text {arctanh}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right ) (2 b c-a d)}{b^{3/2} \sqrt {x}}+\frac {d e \sqrt {a x^2+b x^3}}{b \sqrt {e x}}\) |
Input:
Int[(Sqrt[e*x]*(c + d*x))/Sqrt[a*x^2 + b*x^3],x]
Output:
(d*e*Sqrt[a*x^2 + b*x^3])/(b*Sqrt[e*x]) + ((2*b*c - a*d)*Sqrt[e*x]*ArcTanh [(Sqrt[b]*x^(3/2))/Sqrt[a*x^2 + b*x^3]])/(b^(3/2)*Sqrt[x])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a*x^j + b *x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {d \,x^{2} \left (b x +a \right ) e}{b \sqrt {x^{2} \left (b x +a \right )}\, \sqrt {e x}}-\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {1}{2} a e +b e x}{\sqrt {b e}}+\sqrt {b e \,x^{2}+a e x}\right ) e x \sqrt {e x \left (b x +a \right )}}{2 b \sqrt {b e}\, \sqrt {x^{2} \left (b x +a \right )}\, \sqrt {e x}}\) | \(108\) |
default | \(\frac {x \left (b x +a \right ) \sqrt {e x}\, \left (-\ln \left (\frac {2 b e x +2 \sqrt {e x \left (b x +a \right )}\, \sqrt {b e}+a e}{2 \sqrt {b e}}\right ) a d e +2 \ln \left (\frac {2 b e x +2 \sqrt {e x \left (b x +a \right )}\, \sqrt {b e}+a e}{2 \sqrt {b e}}\right ) b c e +2 \sqrt {b e}\, \sqrt {e x \left (b x +a \right )}\, d \right )}{2 \sqrt {b \,x^{3}+a \,x^{2}}\, \sqrt {e x \left (b x +a \right )}\, b \sqrt {b e}}\) | \(142\) |
Input:
int((e*x)^(1/2)*(d*x+c)/(b*x^3+a*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
d/b*x^2*(b*x+a)*e/(x^2*(b*x+a))^(1/2)/(e*x)^(1/2)-1/2*(a*d-2*b*c)/b*ln((1/ 2*a*e+b*e*x)/(b*e)^(1/2)+(b*e*x^2+a*e*x)^(1/2))/(b*e)^(1/2)*e/(x^2*(b*x+a) )^(1/2)*x*(e*x*(b*x+a))^(1/2)/(e*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\left [-\frac {{\left (2 \, b c - a d\right )} x \sqrt {\frac {e}{b}} \log \left (\frac {2 \, b e x^{2} + a e x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {e x} b \sqrt {\frac {e}{b}}}{x}\right ) - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {e x} d}{2 \, b x}, -\frac {{\left (2 \, b c - a d\right )} x \sqrt {-\frac {e}{b}} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {e x} b \sqrt {-\frac {e}{b}}}{b e x^{2} + a e x}\right ) - \sqrt {b x^{3} + a x^{2}} \sqrt {e x} d}{b x}\right ] \] Input:
integrate((e*x)^(1/2)*(d*x+c)/(b*x^3+a*x^2)^(1/2),x, algorithm="fricas")
Output:
[-1/2*((2*b*c - a*d)*x*sqrt(e/b)*log((2*b*e*x^2 + a*e*x - 2*sqrt(b*x^3 + a *x^2)*sqrt(e*x)*b*sqrt(e/b))/x) - 2*sqrt(b*x^3 + a*x^2)*sqrt(e*x)*d)/(b*x) , -((2*b*c - a*d)*x*sqrt(-e/b)*arctan(sqrt(b*x^3 + a*x^2)*sqrt(e*x)*b*sqrt (-e/b)/(b*e*x^2 + a*e*x)) - sqrt(b*x^3 + a*x^2)*sqrt(e*x)*d)/(b*x)]
\[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\int \frac {\sqrt {e x} \left (c + d x\right )}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \] Input:
integrate((e*x)**(1/2)*(d*x+c)/(b*x**3+a*x**2)**(1/2),x)
Output:
Integral(sqrt(e*x)*(c + d*x)/sqrt(x**2*(a + b*x)), x)
\[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\int { \frac {{\left (d x + c\right )} \sqrt {e x}}{\sqrt {b x^{3} + a x^{2}}} \,d x } \] Input:
integrate((e*x)^(1/2)*(d*x+c)/(b*x^3+a*x^2)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)*sqrt(e*x)/sqrt(b*x^3 + a*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (67) = 134\).
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=-\frac {e^{2} {\left (\frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -\sqrt {b e} \sqrt {e x} + \sqrt {b e^{2} x + a e^{2}} \right |}\right )}{\sqrt {b e} b} - \frac {\sqrt {b e^{2} x + a e^{2}} \sqrt {e x} d}{b e^{2}}\right )}}{{\left | e \right |} \mathrm {sgn}\left (x\right )} + \frac {{\left (2 \, b c e^{2} \log \left (e^{2} {\left | a \right |}\right ) - a d e^{2} \log \left (e^{2} {\left | a \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{2 \, \sqrt {b e} b {\left | e \right |}} \] Input:
integrate((e*x)^(1/2)*(d*x+c)/(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Output:
-e^2*((2*b*c - a*d)*log(abs(-sqrt(b*e)*sqrt(e*x) + sqrt(b*e^2*x + a*e^2))) /(sqrt(b*e)*b) - sqrt(b*e^2*x + a*e^2)*sqrt(e*x)*d/(b*e^2))/(abs(e)*sgn(x) ) + 1/2*(2*b*c*e^2*log(e^2*abs(a)) - a*d*e^2*log(e^2*abs(a)))*sgn(x)/(sqrt (b*e)*b*abs(e))
Timed out. \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\int \frac {\sqrt {e\,x}\,\left (c+d\,x\right )}{\sqrt {b\,x^3+a\,x^2}} \,d x \] Input:
int(((e*x)^(1/2)*(c + d*x))/(a*x^2 + b*x^3)^(1/2),x)
Output:
int(((e*x)^(1/2)*(c + d*x))/(a*x^2 + b*x^3)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {e x} (c+d x)}{\sqrt {a x^2+b x^3}} \, dx=\frac {\sqrt {e}\, \left (\sqrt {x}\, \sqrt {b x +a}\, b d -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a d +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b c \right )}{b^{2}} \] Input:
int((e*x)^(1/2)*(d*x+c)/(b*x^3+a*x^2)^(1/2),x)
Output:
(sqrt(e)*(sqrt(x)*sqrt(a + b*x)*b*d - sqrt(b)*log((sqrt(a + b*x) + sqrt(x) *sqrt(b))/sqrt(a))*a*d + 2*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/s qrt(a))*b*c))/b**2