Integrand size = 26, antiderivative size = 314 \[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2 \sqrt {c+d x}}{a e \sqrt {e x}}-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \left (\sqrt {-a} b c^2+2 a \sqrt {b} c d+(-a)^{3/2} d^2\right ) \arctan \left (\frac {a \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{(-a)^{5/4} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} b e^{3/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b e^{3/2}}-\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+2 \sqrt {-a} \sqrt {b} c d-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} b e^{3/2}} \] Output:
-2*c^2*(d*x+c)^(1/2)/a/e/(e*x)^(1/2)-(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*((-a)^ (1/2)*b*c^2+2*a*b^(1/2)*c*d+(-a)^(3/2)*d^2)*arctan(a*(b^(1/2)*c-(-a)^(1/2) *d)^(1/2)*(e*x)^(1/2)/(-a)^(5/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(7/4)/b/e^(3/ 2)+2*d^(5/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/b/e^(3/2)- (b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(b*c^2+2*(-a)^(1/2)*b^(1/2)*c*d-a*d^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)^(5/4)/b/e^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.49 \[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-2 \left (b c^2 \sqrt {c+d x}+a d^{5/2} \sqrt {x} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )-\sqrt {d} \sqrt {x} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b^2 c^5 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-3 a b c^3 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b^2 c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-6 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+4 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2-3 a b c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{a b (e x)^{3/2}} \] Input:
Integrate[(c + d*x)^(5/2)/((e*x)^(3/2)*(a + b*x^2)),x]
Output:
(x*(-2*(b*c^2*Sqrt[c + d*x] + a*d^(5/2)*Sqrt[x]*Log[-(Sqrt[d]*Sqrt[x]) + S qrt[c + d*x]]) - Sqrt[d]*Sqrt[x]*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 & , (b^2*c^5*Log[c + 2*d*x - 2*Sqrt [d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 3*a*b*c^3*d^2*Log[c + 2*d*x - 2*Sqrt[d]* Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b^2*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]* Sqrt[c + d*x] - #1]*#1 - 6*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*S qrt[c + d*x] - #1]*#1 + 4*a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b^2*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 - 3*a*b*c*d^2*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) & ]))/ (a*b*(e*x)^(3/2))
Time = 1.63 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.09, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \frac {\int \left (\frac {e c^3}{a (e x)^{3/2} \sqrt {c+d x}}+\frac {d^3}{b \sqrt {e x} \sqrt {c+d x}}+\frac {a d \left (3 b c^2-a d^2\right )-b c \left (b c^2-3 a d^2\right ) x}{a b \sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (\sqrt {b} c \left (b c^2-3 a d^2\right )-\sqrt {-a} d \left (3 b c^2-a d^2\right )\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} b \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (3 \sqrt {-a} b c^2 d-3 a \sqrt {b} c d^2+(-a)^{3/2} d^3+b^{3/2} c^3\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} b \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {2 c^2 \sqrt {c+d x}}{a \sqrt {e x}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b \sqrt {e}}}{e}\) |
Input:
Int[(c + d*x)^(5/2)/((e*x)^(3/2)*(a + b*x^2)),x]
Output:
((-2*c^2*Sqrt[c + d*x])/(a*Sqrt[e*x]) + ((Sqrt[b]*c*(b*c^2 - 3*a*d^2) - Sq rt[-a]*d*(3*b*c^2 - a*d^2))*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x] )/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(5/4)*b*Sqrt[Sqrt[b]*c - Sqrt [-a]*d]*Sqrt[e]) + (2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(b*Sqrt[e]) - ((b^(3/2)*c^3 + 3*Sqrt[-a]*b*c^2*d - 3*a*Sqrt[b]*c *d^2 + (-a)^(3/2)*d^3)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/(( -a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(5/4)*b*Sqrt[Sqrt[b]*c + Sqrt[-a] *d]*Sqrt[e]))/e
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. \(560\) vs. \(2(240)=480\).
Time = 0.48 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.79
| method | result | size |
| risch | \(-\frac {2 c^{2} \sqrt {d x +c}}{a e \sqrt {e x}}+\frac {\left (\frac {a \,d^{3} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{b \sqrt {d e}}-\frac {\left (3 \sqrt {-a b}\, a c \,d^{2}-\sqrt {-a b}\, b \,c^{3}-a^{2} d^{3}+3 a b \,c^{2} d \right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {\left (3 \sqrt {-a b}\, a c \,d^{2}-\sqrt {-a b}\, b \,c^{3}+a^{2} d^{3}-3 a b \,c^{2} d \right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a e \sqrt {e x}\, \sqrt {d x +c}}\) | \(561\) |
| default | \(\text {Expression too large to display}\) | \(1132\) |
Input:
int((d*x+c)^(5/2)/(e*x)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2*c^2*(d*x+c)^(1/2)/a/e/(e*x)^(1/2)+1/a*(a/b*d^3*ln((1/2*c*e+d*e*x)/(d*e) ^(1/2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)-1/2*(3*(-a*b)^(1/2)*a*c*d^2-(-a* b)^(1/2)*b*c^3-a^2*d^3+3*a*b*c^2*d)/(-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*(3*(-a*b)^(1/2)*a*c*d^2-(-a*b)^(1/2)*b*c^3 +a^2*d^3-3*a*b*c^2*d)/(-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*((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. 1853 vs. \(2 (240) = 480\).
Time = 2.58 (sec) , antiderivative size = 3712, normalized size of antiderivative = 11.82 \[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**(5/2)/(e*x)**(3/2)/(b*x**2+a),x)
Output:
Integral((c + d*x)**(5/2)/((e*x)**(3/2)*(a + b*x**2)), x)
\[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(5/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)/((b*x^2 + a)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(5/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{{\left (e\,x\right )}^{3/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^(5/2)/((e*x)^(3/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^(5/2)/((e*x)^(3/2)*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^{5/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a x +\sqrt {x}\, b \,x^{3}}d x \right ) c^{2}+2 \left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a +\sqrt {x}\, b \,x^{2}}d x \right ) c d +\left (\int \frac {\sqrt {d x +c}\, x}{\sqrt {x}\, a +\sqrt {x}\, b \,x^{2}}d x \right ) d^{2}}{\sqrt {e}\, e} \] Input:
int((d*x+c)^(5/2)/(e*x)^(3/2)/(b*x^2+a),x)
Output:
(int(sqrt(c + d*x)/(sqrt(x)*a*x + sqrt(x)*b*x**3),x)*c**2 + 2*int(sqrt(c + d*x)/(sqrt(x)*a + sqrt(x)*b*x**2),x)*c*d + int((sqrt(c + d*x)*x)/(sqrt(x) *a + sqrt(x)*b*x**2),x)*d**2)/(sqrt(e)*e)