Integrand size = 26, antiderivative size = 209 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 c \sqrt {c+d x}}{a e \sqrt {e x}}+\frac {\left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {b} e^{3/2}}-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {b} e^{3/2}} \] Output:
-2*c*(d*x+c)^(1/2)/a/e/(e*x)^(1/2)+(b^(1/2)*c-(-a)^(1/2)*d)^(3/2)*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)^(5/4)/b^(1/2)/e^(3/2)-(b^(1/2)*c+(-a)^(1/2)*d)^(3/2)*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 )^(5/4)/b^(1/2)/e^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.31 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-4 c \sqrt {c+d x}+c \sqrt {x} \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {a d^4 \log (x)-2 a d^4 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-2 b c^2 d \log (x) \text {$\#$1}^2-a d^3 \log (x) \text {$\#$1}^2+4 b c^2 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+2 a d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+2 b c^2 \log (x) \text {$\#$1}^4+a d^2 \log (x) \text {$\#$1}^4-4 b c^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 a d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-a d \log (x) \text {$\#$1}^6+2 a d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{a d^3 \text {$\#$1}-8 b c^2 \text {$\#$1}^3-3 a d^2 \text {$\#$1}^3+3 a d \text {$\#$1}^5-a \text {$\#$1}^7}\&\right ]\right )}{2 a (e x)^{3/2}} \] Input:
Integrate[(c + d*x)^(3/2)/((e*x)^(3/2)*(a + b*x^2)),x]
Output:
(x*(-4*c*Sqrt[c + d*x] + c*Sqrt[x]*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 & , (a*d^4*Log[x] - 2*a*d^4*Log [-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 2*b*c^2*d*Log[x]*#1^2 - a*d^3*Lo g[x]*#1^2 + 4*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 2* a*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 2*b*c^2*Log[x]*#1^ 4 + a*d^2*Log[x]*#1^4 - 4*b*c^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] *#1^4 - 2*a*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - a*d*Log[ x]*#1^6 + 2*a*d*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) & ]))/(2*a*(e*x)^(3/ 2))
Time = 1.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.27, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \frac {\int \left (\frac {e c^2}{a (e x)^{3/2} \sqrt {c+d x}}+\frac {2 a c d-\left (b c^2-a d^2\right ) x}{a \sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {b} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {b} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {2 c \sqrt {c+d x}}{a \sqrt {e x}}}{e}\) |
Input:
Int[(c + d*x)^(3/2)/((e*x)^(3/2)*(a + b*x^2)),x]
Output:
((-2*c*Sqrt[c + d*x])/(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)^(5/4)*Sqrt[b]*Sqrt[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 + Sqr t[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(5/4)*Sqrt[ 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. \(492\) vs. \(2(153)=306\).
Time = 0.49 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.36
| method | result | size |
| risch | \(-\frac {2 c \sqrt {d x +c}}{a e \sqrt {e x}}+\frac {\left (-\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}-2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}-\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}+2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a e \sqrt {e x}\, \sqrt {d x +c}}\) | \(493\) |
| default | \(-\frac {\sqrt {d x +c}\, \left (2 \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 b c d e x +\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 ) a \,d^{2} e x -\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 ) b \,c^{2} e x -2 \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 b c d e x +\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}}\, a \,d^{2} e x -\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}}\, b \,c^{2} e x +4 b c \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\right )}{2 e \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a \sqrt {-a b}\, 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}}}\) | \(778\) |
Input:
int((d*x+c)^(3/2)/(e*x)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2*c*(d*x+c)^(1/2)/a/e/(e*x)^(1/2)+1/a*(-1/2*((-a*b)^(1/2)*a*d^2-(-a*b)^(1 /2)*b*c^2-2*a*b*c*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*((-a*b)^(1/2)*a*d^2-(-a*b)^(1/2)*b*c^2+2*a*b*c*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. 1173 vs. \(2 (153) = 306\).
Time = 0.13 (sec) , antiderivative size = 1173, normalized size of antiderivative = 5.61 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="fricas")
Output:
1/2*(a*e^2*x*sqrt((a^2*b*e^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^ 4)/(a^5*b*e^6)) + 3*b*c^2*d - a*d^3)/(a^2*b*e^3))*log(-((b^2*c^5 - 2*a*b*c ^3*d^2 - 3*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + (a^4*b*d*e^5*x*sqrt(-(b^2* c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) + (a*b^2*c^4 - 3*a^2*b*c ^2*d^2)*e^2*x)*sqrt((a^2*b*e^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2* d^4)/(a^5*b*e^6)) + 3*b*c^2*d - a*d^3)/(a^2*b*e^3)))/x) - a*e^2*x*sqrt((a^ 2*b*e^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) + 3*b *c^2*d - a*d^3)/(a^2*b*e^3))*log(-((b^2*c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4) *sqrt(d*x + c)*sqrt(e*x) - (a^4*b*d*e^5*x*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) + (a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e^2*x)*sqrt(( a^2*b*e^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) + 3 *b*c^2*d - a*d^3)/(a^2*b*e^3)))/x) - a*e^2*x*sqrt(-(a^2*b*e^3*sqrt(-(b^2*c ^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) - 3*b*c^2*d + a*d^3)/(a^2 *b*e^3))*log(-((b^2*c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4)*sqrt(d*x + c)*sqrt( e*x) + (a^4*b*d*e^5*x*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5 *b*e^6)) - (a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e^2*x)*sqrt(-(a^2*b*e^3*sqrt(-(b^ 2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) - 3*b*c^2*d + a*d^3)/( a^2*b*e^3)))/x) + a*e^2*x*sqrt(-(a^2*b*e^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b*e^6)) - 3*b*c^2*d + a*d^3)/(a^2*b*e^3))*log(-((b^2 *c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) - (a^4*b*d*...
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**(3/2)/(e*x)**(3/2)/(b*x**2+a),x)
Output:
Integral((c + d*x)**(3/2)/((e*x)**(3/2)*(a + b*x**2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((b*x^2 + a)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^(3/2)/((e*x)^(3/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^(3/2)/((e*x)^(3/2)*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a x +\sqrt {x}\, b \,x^{3}}d x \right ) c +\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a +\sqrt {x}\, b \,x^{2}}d x \right ) d}{\sqrt {e}\, e} \] Input:
int((d*x+c)^(3/2)/(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 + int(sqrt(c + d*x) /(sqrt(x)*a + sqrt(x)*b*x**2),x)*d)/(sqrt(e)*e)