Integrand size = 26, antiderivative size = 270 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 c \sqrt {c+d x}}{3 a e (e x)^{3/2}}-\frac {8 d \sqrt {c+d x}}{3 a e^2 \sqrt {e x}}-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \left (\sqrt [4]{-a} \sqrt {b} c-(-a)^{3/4} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{a^2 e^{5/2}}-\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \left (\sqrt [4]{-a} \sqrt {b} c+(-a)^{3/4} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{a^2 e^{5/2}} \] Output:
-2/3*c*(d*x+c)^(1/2)/a/e/(e*x)^(3/2)-8/3*d*(d*x+c)^(1/2)/a/e^2/(e*x)^(1/2) -(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*((-a)^(1/4)*b^(1/2)*c-(-a)^(3/4)*d)*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^2/e^(5/2)-(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*((-a)^(1/4)*b^(1/2)*c+(-a)^ (3/4)*d)*arctanh((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^2/e^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.96 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-8 \sqrt {c+d x} (c+4 d x)-3 x^{3/2} \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-b c^2 d^3 \log (x)+a d^5 \log (x)+2 b c^2 d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-2 a d^5 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-5 b c^2 d^2 \log (x) \text {$\#$1}^2-3 a d^4 \log (x) \text {$\#$1}^2+10 b c^2 d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+6 a d^4 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+5 b c^2 d \log (x) \text {$\#$1}^4+3 a d^3 \log (x) \text {$\#$1}^4-10 b c^2 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-6 a d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+b c^2 \log (x) \text {$\#$1}^6-a d^2 \log (x) \text {$\#$1}^6-2 b c^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6+2 a d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{-a d^3 \text {$\#$1}+8 b c^2 \text {$\#$1}^3+3 a d^2 \text {$\#$1}^3-3 a d \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]\right )}{12 a (e x)^{5/2}} \] Input:
Integrate[(c + d*x)^(3/2)/((e*x)^(5/2)*(a + b*x^2)),x]
Output:
(x*(-8*Sqrt[c + d*x]*(c + 4*d*x) - 3*x^(3/2)*RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b*c^2*#1^4 + 6*a*d^2*#1^4 - 4*a*d*#1^6 + a*#1^8 & , (-(b*c^2*d^3*Log[ x]) + a*d^5*Log[x] + 2*b*c^2*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1 ] - 2*a*d^5*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 5*b*c^2*d^2*Log[x ]*#1^2 - 3*a*d^4*Log[x]*#1^2 + 10*b*c^2*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 6*a*d^4*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^ 2 + 5*b*c^2*d*Log[x]*#1^4 + 3*a*d^3*Log[x]*#1^4 - 10*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - 6*a*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 + b*c^2*Log[x]*#1^6 - a*d^2*Log[x]*#1^6 - 2*b*c^2*Log[- Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6 + 2*a*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6)/(-(a*d^3*#1) + 8*b*c^2*#1^3 + 3*a*d^2*#1^3 - 3 *a*d*#1^5 + a*#1^7) & ]))/(12*a*(e*x)^(5/2))
Time = 1.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \frac {\int \left (\frac {c^2 e^2}{a (e x)^{5/2} \sqrt {c+d x}}+\frac {2 c d e}{a (e x)^{3/2} \sqrt {c+d x}}-\frac {b c^2+2 b d x c-a d^2}{a \sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {8 d \sqrt {c+d x}}{3 a \sqrt {e x}}-\frac {2 c e \sqrt {c+d x}}{3 a (e x)^{3/2}}}{e^2}\) |
Input:
Int[(c + d*x)^(3/2)/((e*x)^(5/2)*(a + b*x^2)),x]
Output:
((-2*c*e*Sqrt[c + d*x])/(3*a*(e*x)^(3/2)) - (8*d*Sqrt[c + d*x])/(3*a*Sqrt[ e*x]) - ((b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(7/4)*S qrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) - ((b*c^2 + 2*Sqrt[-a]*Sqrt[b]*c*d - a*d^2)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e ]*Sqrt[c + d*x])])/((-a)^(7/4)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*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. \(479\) vs. \(2(202)=404\).
Time = 0.46 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(-\frac {2 \sqrt {d x +c}\, \left (4 d x +c \right )}{3 a x \,e^{2} \sqrt {e x}}-\frac {\left (-\frac {\left (-a \,d^{2}+b \,c^{2}+2 c d \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {\left (a \,d^{2}-b \,c^{2}+2 c d \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a \,e^{2} \sqrt {e x}\, \sqrt {d x +c}}\) | \(480\) |
| default | \(\frac {\sqrt {d x +c}\, \left (6 \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 b c d e \,x^{2} \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+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 ) a \,d^{2} e \,x^{2} \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}-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 ) b \,c^{2} e \,x^{2} \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+6 \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 b c d e \,x^{2} \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}-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 ) a \,d^{2} e \,x^{2} \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}+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 ) b \,c^{2} e \,x^{2} \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}-16 x \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, a b d -4 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, a b c \right )}{6 e^{2} x \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}}\) | \(840\) |
Input:
int((d*x+c)^(3/2)/(e*x)^(5/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/3*(d*x+c)^(1/2)*(4*d*x+c)/a/x/e^2/(e*x)^(1/2)-1/a*(-1/2*(-a*d^2+b*c^2+2 *c*d*(-a*b)^(1/2))/(-a*b)^(1/2)/(-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*d^2-b*c^2+2*c*d*(-a*b)^(1/2))/(-a*b)^(1/2)/(-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. 1132 vs. \(2 (202) = 404\).
Time = 0.12 (sec) , antiderivative size = 1132, normalized size of antiderivative = 4.19 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/6*(3*a*e^3*x^2*sqrt(-(a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2* b*c^2*d^4)/(a^7*e^10)) + 3*b*c^2*d - a*d^3)/(a^3*e^5))*log(-((b^2*c^4 - 2* a*b*c^2*d^2 - 3*a^2*d^4)*sqrt(d*x + c)*sqrt(e*x) + (a^5*e^8*x*sqrt(-(b^3*c ^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10)) - (a^2*b*c^2*d - 3*a^3 *d^3)*e^3*x)*sqrt(-(a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2 *d^4)/(a^7*e^10)) + 3*b*c^2*d - a*d^3)/(a^3*e^5)))/x) - 3*a*e^3*x^2*sqrt(- (a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10)) + 3*b*c^2*d - a*d^3)/(a^3*e^5))*log(-((b^2*c^4 - 2*a*b*c^2*d^2 - 3*a^2*d^4) *sqrt(d*x + c)*sqrt(e*x) - (a^5*e^8*x*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9 *a^2*b*c^2*d^4)/(a^7*e^10)) - (a^2*b*c^2*d - 3*a^3*d^3)*e^3*x)*sqrt(-(a^3* e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10)) + 3*b* c^2*d - a*d^3)/(a^3*e^5)))/x) - 3*a*e^3*x^2*sqrt((a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10)) - 3*b*c^2*d + a*d^3)/(a^3* e^5))*log(-((b^2*c^4 - 2*a*b*c^2*d^2 - 3*a^2*d^4)*sqrt(d*x + c)*sqrt(e*x) + (a^5*e^8*x*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10 )) + (a^2*b*c^2*d - 3*a^3*d^3)*e^3*x)*sqrt((a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b ^2*c^4*d^2 + 9*a^2*b*c^2*d^4)/(a^7*e^10)) - 3*b*c^2*d + a*d^3)/(a^3*e^5))) /x) + 3*a*e^3*x^2*sqrt((a^3*e^5*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b *c^2*d^4)/(a^7*e^10)) - 3*b*c^2*d + a*d^3)/(a^3*e^5))*log(-((b^2*c^4 - 2*a *b*c^2*d^2 - 3*a^2*d^4)*sqrt(d*x + c)*sqrt(e*x) - (a^5*e^8*x*sqrt(-(b^3...
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**(3/2)/(e*x)**(5/2)/(b*x**2+a),x)
Output:
Integral((c + d*x)**(3/2)/((e*x)**(5/2)*(a + b*x**2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((b*x^2 + a)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(5/2)/(b*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^(3/2)/((e*x)^(5/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^(3/2)/((e*x)^(5/2)*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{4}}d x \right ) c +\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a x +\sqrt {x}\, b \,x^{3}}d x \right ) d}{\sqrt {e}\, e^{2}} \] Input:
int((d*x+c)^(3/2)/(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)*c + int(sqrt(c + d *x)/(sqrt(x)*a*x + sqrt(x)*b*x**3),x)*d)/(sqrt(e)*e**2)