Integrand size = 26, antiderivative size = 214 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 (c+d x)^{3/2}}{3 a c e (e x)^{3/2}}-\frac {\sqrt {b} \sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2}}-\frac {\sqrt {b} \sqrt {\sqrt {b} c+\sqrt {-a} d} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2}} \] Output:
-2/3*(d*x+c)^(3/2)/a/c/e/(e*x)^(3/2)-b^(1/2)*(b^(1/2)*c-(-a)^(1/2)*d)^(1/2 )*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d* x+c)^(1/2))/(-a)^(7/4)/e^(5/2)-b^(1/2)*(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*arct anh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^ (1/2))/(-a)^(7/4)/e^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-2 (c+d x)^{3/2}-3 b c d^{3/2} x^{3/2} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+2 c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{3 a c (e x)^{5/2}} \] Input:
Integrate[Sqrt[c + d*x]/((e*x)^(5/2)*(a + b*x^2)),x]
Output:
(x*(-2*(c + d*x)^(3/2) - 3*b*c*d^(3/2)*x^(3/2)*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , (c^2*Log[c + 2*d* x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 2*c*Log[c + 2*d*x - 2*Sqrt[d]* Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^3 - 3*b*c^2*#1 - 8*a*d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ]))/(3*a*c*(e*x)^(5/2))
Time = 0.90 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {610, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \frac {\int \left (\frac {c e^2}{a (e x)^{5/2} \sqrt {c+d x}}+\frac {d e}{a (e x)^{3/2} \sqrt {c+d x}}-\frac {b \sqrt {c+d x}}{a \sqrt {e x} \left (b x^2+a\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt {b} \sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {e}}-\frac {\sqrt {b} \sqrt {\sqrt {-a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {e}}-\frac {2 d \sqrt {c+d x}}{3 a c \sqrt {e x}}-\frac {2 e \sqrt {c+d x}}{3 a (e x)^{3/2}}}{e^2}\) |
Input:
Int[Sqrt[c + d*x]/((e*x)^(5/2)*(a + b*x^2)),x]
Output:
((-2*e*Sqrt[c + d*x])/(3*a*(e*x)^(3/2)) - (2*d*Sqrt[c + d*x])/(3*a*c*Sqrt[ e*x]) - (Sqrt[b]*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*ArcTan[(Sqrt[Sqrt[b]*c - Sqr t[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(7/4)*Sqrt[ e]) - (Sqrt[b]*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt [-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(7/4)*Sqrt[e ]))/e^2
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[e^(m + 1/2) Int[ExpandIntegrand[1/(Sqrt[e*x]*Sqrt[c + d*x] ), x^(m + 1/2)*((c + d*x)^(n + 1/2)/(a + b*x^2)), x], x], x] /; FreeQ[{a, b , c, d, e}, x] && IGtQ[n + 1/2, 0] && ILtQ[m - 1/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(464\) vs. \(2(156)=312\).
Time = 0.46 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.17
| method | result | size |
| risch | \(-\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 a x c \,e^{2} \sqrt {e x}}-\frac {b \left (-\frac {\left (\sqrt {-a b}\, d -b c \right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}-\frac {\left (\sqrt {-a b}\, d +b c \right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a \,e^{2} \sqrt {e x}\, \sqrt {d x +c}}\) | \(465\) |
| default | \(\frac {\sqrt {d x +c}\, \left (3 \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a c d e \,x^{2}+3 \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) c^{2} e \,x^{2}+3 \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a c d e \,x^{2}-3 \sqrt {-a b}\, \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, c^{2} e \,x^{2}-4 a d x \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}-4 a c \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\right )}{6 e^{2} x \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\) | \(613\) |
Input:
int((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/3*(d*x+c)^(3/2)/a/x/c/e^2/(e*x)^(1/2)-1/a*b*(-1/2*((-a*b)^(1/2)*d-b*c)/ (-a*b)^(1/2)/b/(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d+c*(-a*b)^(1 /2))/b+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1/2)/b)+2*(-e*(a*d+c*(-a*b)^ (1/2))/b)^(1/2)*(d*e*(x+(-a*b)^(1/2)/b)^2+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+( -a*b)^(1/2)/b)-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2))/(x+(-a*b)^(1/2)/b))-1/2*(( -a*b)^(1/2)*d+b*c)/(-a*b)^(1/2)/b/(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*ln((-2 *e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)+2* (-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x-(-a*b)^(1/2)/b)^2+e*(2*(-a*b)^(1 /2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2))/(x-(-a*b) ^(1/2)/b)))/e^2*((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (156) = 312\).
Time = 0.12 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {3 \, a c e^{3} x^{2} \sqrt {-\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + b d}{a^{3} e^{5}}} \log \left (\frac {a^{5} e^{8} x \sqrt {-\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + b d}{a^{3} e^{5}}} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + \sqrt {d x + c} \sqrt {e x} b^{2} c}{x}\right ) - 3 \, a c e^{3} x^{2} \sqrt {-\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + b d}{a^{3} e^{5}}} \log \left (-\frac {a^{5} e^{8} x \sqrt {-\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + b d}{a^{3} e^{5}}} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - \sqrt {d x + c} \sqrt {e x} b^{2} c}{x}\right ) - 3 \, a c e^{3} x^{2} \sqrt {\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - b d}{a^{3} e^{5}}} \log \left (\frac {a^{5} e^{8} x \sqrt {\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - b d}{a^{3} e^{5}}} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} + \sqrt {d x + c} \sqrt {e x} b^{2} c}{x}\right ) + 3 \, a c e^{3} x^{2} \sqrt {\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - b d}{a^{3} e^{5}}} \log \left (-\frac {a^{5} e^{8} x \sqrt {\frac {a^{3} e^{5} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - b d}{a^{3} e^{5}}} \sqrt {-\frac {b^{3} c^{2}}{a^{7} e^{10}}} - \sqrt {d x + c} \sqrt {e x} b^{2} c}{x}\right ) + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {e x}}{6 \, a c e^{3} x^{2}} \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/6*(3*a*c*e^3*x^2*sqrt(-(a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) + b*d)/(a^3*e ^5))*log((a^5*e^8*x*sqrt(-(a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) + b*d)/(a^3*e ^5))*sqrt(-b^3*c^2/(a^7*e^10)) + sqrt(d*x + c)*sqrt(e*x)*b^2*c)/x) - 3*a*c *e^3*x^2*sqrt(-(a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) + b*d)/(a^3*e^5))*log(-( a^5*e^8*x*sqrt(-(a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) + b*d)/(a^3*e^5))*sqrt( -b^3*c^2/(a^7*e^10)) - sqrt(d*x + c)*sqrt(e*x)*b^2*c)/x) - 3*a*c*e^3*x^2*s qrt((a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) - b*d)/(a^3*e^5))*log((a^5*e^8*x*sq rt((a^3*e^5*sqrt(-b^3*c^2/(a^7*e^10)) - b*d)/(a^3*e^5))*sqrt(-b^3*c^2/(a^7 *e^10)) + sqrt(d*x + c)*sqrt(e*x)*b^2*c)/x) + 3*a*c*e^3*x^2*sqrt((a^3*e^5* sqrt(-b^3*c^2/(a^7*e^10)) - b*d)/(a^3*e^5))*log(-(a^5*e^8*x*sqrt((a^3*e^5* sqrt(-b^3*c^2/(a^7*e^10)) - b*d)/(a^3*e^5))*sqrt(-b^3*c^2/(a^7*e^10)) - sq rt(d*x + c)*sqrt(e*x)*b^2*c)/x) + 4*(d*x + c)^(3/2)*sqrt(e*x))/(a*c*e^3*x^ 2)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c + d x}}{\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**(1/2)/(e*x)**(5/2)/(b*x**2+a),x)
Output:
Integral(sqrt(c + d*x)/((e*x)**(5/2)*(a + b*x**2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/((b*x^2 + a)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (e\,x\right )}^{5/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^(1/2)/((e*x)^(5/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^(1/2)/((e*x)^(5/2)*(a + b*x^2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{4}}d x}{\sqrt {e}\, e^{2}} \] Input:
int((d*x+c)^(1/2)/(e*x)^(5/2)/(b*x^2+a),x)
Output:
int(sqrt(c + d*x)/(sqrt(x)*a*x**2 + sqrt(x)*b*x**4),x)/(sqrt(e)*e**2)