Integrand size = 26, antiderivative size = 280 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-\frac {2 c \sqrt {c+d x}}{5 a e (e x)^{5/2}}-\frac {4 d \sqrt {c+d x}}{5 a e^2 (e x)^{3/2}}+\frac {2 \left (5 b c^2-a d^2\right ) \sqrt {c+d x}}{5 a^2 c e^3 \sqrt {e x}}+\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} e^{7/2}}-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} e^{7/2}} \] Output:
-2/5*c*(d*x+c)^(1/2)/a/e/(e*x)^(5/2)-4/5*d*(d*x+c)^(1/2)/a/e^2/(e*x)^(3/2) +2/5*(-a*d^2+5*b*c^2)*(d*x+c)^(1/2)/a^2/c/e^3/(e*x)^(1/2)+b^(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)^(9/4)/e^(7/2)-b^(1/2)*(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)^(9/4)/e^(7/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.41 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\frac {x \left (2 \sqrt {c+d x} \left (5 b c^2 x^2-a (c+d x)^2\right )+5 b c \sqrt {d} x^{5/2} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-a c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-6 a c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2-a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{5 a^2 c (e x)^{7/2}} \] Input:
Integrate[(c + d*x)^(3/2)/((e*x)^(7/2)*(a + b*x^2)),x]
Output:
(x*(2*Sqrt[c + d*x]*(5*b*c^2*x^2 - a*(c + d*x)^2) + 5*b*c*Sqrt[d]*x^(5/2)* 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 & , (b*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - a *c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b*c^3*L og[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 6*a*c*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b*c^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 - a*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) & ]))/(5*a^2*c*(e*x)^(7/2))
Time = 1.41 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.29, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 610 |
| \(\displaystyle \frac {\int \left (\frac {c^2 e^3}{a (e x)^{7/2} \sqrt {c+d x}}+\frac {2 c d e^2}{a (e x)^{5/2} \sqrt {c+d x}}-\frac {\left (b c^2-a d^2\right ) e}{a^2 (e x)^{3/2} \sqrt {c+d x}}-\frac {b \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{a^2 \sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}\right )dx}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} \left (b c^2-a d^2\right )}{a^2 c \sqrt {e x}}+\frac {\sqrt {b} \left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\sqrt {b} \left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}+\frac {8 d^2 \sqrt {c+d x}}{5 a c \sqrt {e x}}-\frac {2 c e^2 \sqrt {c+d x}}{5 a (e x)^{5/2}}-\frac {4 d e \sqrt {c+d x}}{5 a (e x)^{3/2}}}{e^3}\) |
Input:
Int[(c + d*x)^(3/2)/((e*x)^(7/2)*(a + b*x^2)),x]
Output:
((-2*c*e^2*Sqrt[c + d*x])/(5*a*(e*x)^(5/2)) - (4*d*e*Sqrt[c + d*x])/(5*a*( e*x)^(3/2)) + (8*d^2*Sqrt[c + d*x])/(5*a*c*Sqrt[e*x]) + (2*(b*c^2 - a*d^2) *Sqrt[c + d*x])/(a^2*c*Sqrt[e*x]) + (Sqrt[b]*(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)*Sq rt[e]*Sqrt[c + d*x])])/((-a)^(9/4)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) - (Sqrt[b]*(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)^(9/4) *Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e]))/e^3
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. \(528\) vs. \(2(210)=420\).
Time = 0.47 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(-\frac {2 \sqrt {d x +c}\, \left (a \,d^{2} x^{2}-5 b \,c^{2} x^{2}+2 a d x c +a \,c^{2}\right )}{5 c \,a^{2} x^{2} e^{3} \sqrt {e x}}-\frac {b \left (-\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}-2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}-\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}+2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a^{2} e^{3} \sqrt {e x}\, \sqrt {d x +c}}\) | \(529\) |
| default | \(-\frac {\sqrt {d x +c}\, \left (-10 \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^{2} d e \,x^{3}-5 \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 c \,d^{2} e \,x^{3}+5 \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^{3} e \,x^{3}+10 \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^{2} d e \,x^{3}-5 \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 c \,d^{2} e \,x^{3}+5 \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^{3} e \,x^{3}+4 a \,d^{2} x^{2} \sqrt {-a b}\, \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}}-20 b \,c^{2} x^{2} \sqrt {-a b}\, \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}}+8 a c d x \sqrt {-a b}\, \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^{2} \sqrt {-a b}\, \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 )}{10 e^{3} x^{2} c \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} \sqrt {-a 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}}}\) | \(995\) |
Input:
int((d*x+c)^(3/2)/(e*x)^(7/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-2/5*(d*x+c)^(1/2)*(a*d^2*x^2-5*b*c^2*x^2+2*a*c*d*x+a*c^2)/c/a^2/x^2/e^3/( e*x)^(1/2)-1/a^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))-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^3*((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. 1275 vs. \(2 (210) = 420\).
Time = 0.13 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.55 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/10*(5*a^2*c*e^4*x^3*sqrt((a^4*e^7*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9* a^2*b^3*c^2*d^4)/(a^9*e^14)) + 3*b^2*c^2*d - a*b*d^3)/(a^4*e^7))*log(-((b^ 4*c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + (a^7* d*e^11*x*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) + (a^2*b^3*c^4 - 3*a^3*b^2*c^2*d^2)*e^4*x)*sqrt((a^4*e^7*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) + 3*b^2*c^2*d - a*b*d^3) /(a^4*e^7)))/x) - 5*a^2*c*e^4*x^3*sqrt((a^4*e^7*sqrt(-(b^5*c^6 - 6*a*b^4*c ^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) + 3*b^2*c^2*d - a*b*d^3)/(a^4*e^7) )*log(-((b^4*c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e *x) - (a^7*d*e^11*x*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/ (a^9*e^14)) + (a^2*b^3*c^4 - 3*a^3*b^2*c^2*d^2)*e^4*x)*sqrt((a^4*e^7*sqrt( -(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) + 3*b^2*c^2*d - a*b*d^3)/(a^4*e^7)))/x) - 5*a^2*c*e^4*x^3*sqrt(-(a^4*e^7*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) - 3*b^2*c^2*d + a*b*d^ 3)/(a^4*e^7))*log(-((b^4*c^5 - 2*a*b^3*c^3*d^2 - 3*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + (a^7*d*e^11*x*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b ^3*c^2*d^4)/(a^9*e^14)) - (a^2*b^3*c^4 - 3*a^3*b^2*c^2*d^2)*e^4*x)*sqrt(-( a^4*e^7*sqrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) - 3*b^2*c^2*d + a*b*d^3)/(a^4*e^7)))/x) + 5*a^2*c*e^4*x^3*sqrt(-(a^4*e^7*s qrt(-(b^5*c^6 - 6*a*b^4*c^4*d^2 + 9*a^2*b^3*c^2*d^4)/(a^9*e^14)) - 3*b^...
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**(3/2)/(e*x)**(7/2)/(b*x**2+a),x)
Output:
Integral((c + d*x)**(3/2)/((e*x)**(7/2)*(a + b*x**2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)/((b*x^2 + a)*(e*x)^(7/2)), x)
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(3/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}\,\left (b\,x^2+a\right )} \,d x \] Input:
int((c + d*x)^(3/2)/((e*x)^(7/2)*(a + b*x^2)),x)
Output:
int((c + d*x)^(3/2)/((e*x)^(7/2)*(a + b*x^2)), x)
\[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\frac {\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a \,x^{3}+\sqrt {x}\, b \,x^{5}}d x \right ) c +\left (\int \frac {\sqrt {d x +c}}{\sqrt {x}\, a \,x^{2}+\sqrt {x}\, b \,x^{4}}d x \right ) d}{\sqrt {e}\, e^{3}} \] Input:
int((d*x+c)^(3/2)/(e*x)^(7/2)/(b*x^2+a),x)
Output:
(int(sqrt(c + d*x)/(sqrt(x)*a*x**3 + sqrt(x)*b*x**5),x)*c + int(sqrt(c + d *x)/(sqrt(x)*a*x**2 + sqrt(x)*b*x**4),x)*d)/(sqrt(e)*e**3)