Integrand size = 26, antiderivative size = 249 \[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\frac {d \sqrt {e x} \sqrt {c+d x}}{b}+\frac {\left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} \sqrt {e} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} b^{3/2}}+\frac {3 c \sqrt {d} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{b}-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} b^{3/2}} \] Output:
d*(e*x)^(1/2)*(d*x+c)^(1/2)/b+(b^(1/2)*c-(-a)^(1/2)*d)^(3/2)*e^(1/2)*arcta n((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)^(1/4)/b^(3/2)+3*c*d^(1/2)*e^(1/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^ (1/2)/(d*x+c)^(1/2))/b-(b^(1/2)*c+(-a)^(1/2)*d)^(3/2)*e^(1/2)*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)^(1/4)/b^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.47 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\frac {\sqrt {e x} \left (d \sqrt {x} \sqrt {c+d x}+6 c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )-\frac {1}{2} c \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {a d^4 \log (x)-2 a d^4 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-2 b c^2 d \log (x) \text {$\#$1}^2-a d^3 \log (x) \text {$\#$1}^2+4 b c^2 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+2 a d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+2 b c^2 \log (x) \text {$\#$1}^4+a d^2 \log (x) \text {$\#$1}^4-4 b c^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 a d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-a d \log (x) \text {$\#$1}^6+2 a d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{a d^3 \text {$\#$1}-8 b c^2 \text {$\#$1}^3-3 a d^2 \text {$\#$1}^3+3 a d \text {$\#$1}^5-a \text {$\#$1}^7}\&\right ]\right )}{b \sqrt {x}} \] Input:
Integrate[(Sqrt[e*x]*(c + d*x)^(3/2))/(a + b*x^2),x]
Output:
(Sqrt[e*x]*(d*Sqrt[x]*Sqrt[c + d*x] + 6*c*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[x] )/(-Sqrt[c] + Sqrt[c + d*x])] - (c*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))/(b*Sqrt[x])
Time = 1.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {607, 90, 65, 221, 2353, 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.
| \(\displaystyle \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 607 |
| \(\displaystyle \frac {d e \int \frac {2 c+d x}{\sqrt {e x} \sqrt {c+d x}}dx}{b}-\frac {e \int \frac {2 a c d-\left (b c^2-a d^2\right ) x}{\sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {d e \left (\frac {3}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{b}-\frac {e \int \frac {2 a c d-\left (b c^2-a d^2\right ) x}{\sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {d e \left (3 c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{b}-\frac {e \int \frac {2 a c d-\left (b c^2-a d^2\right ) x}{\sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d e \left (\frac {3 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{b}-\frac {e \int \frac {2 a c d-\left (b c^2-a d^2\right ) x}{\sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle \frac {d e \left (\frac {3 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{b}-\frac {e \int \left (\frac {2 \sqrt {-a} a c d-\frac {a \left (a d^2-b c^2\right )}{\sqrt {b}}}{2 a \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}+\frac {2 \sqrt {-a} a c d+\frac {a \left (a d^2-b c^2\right )}{\sqrt {b}}}{2 a \sqrt {e x} \left (\sqrt {b} x+\sqrt {-a}\right ) \sqrt {c+d x}}\right )dx}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d e \left (\frac {3 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{b}-\frac {e \left (\frac {\left (2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \sqrt {b} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {\left (-2 \sqrt {-a} \sqrt {b} c d-a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \sqrt {b} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}\right )}{b}\) |
Input:
Int[(Sqrt[e*x]*(c + d*x)^(3/2))/(a + b*x^2),x]
Output:
(d*e*((Sqrt[e*x]*Sqrt[c + d*x])/e + (3*c*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt [e]*Sqrt[c + d*x])])/(Sqrt[d]*Sqrt[e])))/b - (e*(-(((b*c^2 - 2*Sqrt[-a]*Sq rt[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)^(1/4)*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[S qrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((- a)^(1/4)*Sqrt[b]*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e])))/b
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[e*(d/b) Int[(e*x)^(m - 1)*(c + d*x)^(n - 2)*(2*c + d*x), x ], x] - Simp[e/b Int[(e*x)^(m - 1)*(c + d*x)^(n - 2)*(Simp[2*a*c*d - (b*c ^2 - a*d^2)*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[ n, 1] && GtQ[m, 0] && !IntegerQ[m] && !IntegerQ[n]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(181)=362\).
Time = 0.47 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.12
| method | result | size |
| risch | \(\frac {x \sqrt {d x +c}\, d e}{b \sqrt {e x}}+\frac {\left (\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}+2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}+\frac {\left (\sqrt {-a b}\, a \,d^{2}-\sqrt {-a b}\, b \,c^{2}-2 a b c d \right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}+\frac {3 c d \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}\right ) e \sqrt {\left (d x +c \right ) e x}}{2 b \sqrt {e x}\, \sqrt {d x +c}}\) | \(527\) |
| default | \(\frac {\sqrt {e x}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, b c d e +2 \sqrt {d e}\, \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 +\sqrt {d e}\, \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 ) \sqrt {-a b}\, a \,d^{2} e -\sqrt {d e}\, \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 ) \sqrt {-a b}\, b \,c^{2} e -2 \sqrt {d e}\, \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 +\sqrt {d e}\, \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}}\, \sqrt {-a b}\, a \,d^{2} e -\sqrt {d e}\, \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}}\, \sqrt {-a b}\, b \,c^{2} e +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-a b}\, b d \right )}{2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-a b}\, b^{2} \sqrt {d e}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}}\) | \(892\) |
Input:
int((e*x)^(1/2)*(d*x+c)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/b*x*(d*x+c)^(1/2)*d*e/(e*x)^(1/2)+1/2/b*(((-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))+((-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))+3*c*d*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))*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. 1122 vs. \(2 (181) = 362\).
Time = 0.32 (sec) , antiderivative size = 2243, normalized size of antiderivative = 9.01 \[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(1/2)*(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="fricas")
Output:
[1/2*(3*sqrt(d*e)*c*log(2*d*e*x + c*e + 2*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x )) - b*sqrt((b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^ 5)) + (3*b*c^2*d - a*d^3)*e)/b^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)*e + (a*b^4*d*x*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^ 2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) + (b^3*c^4 - 3*a*b^2*c^2*d^2)*e*x)*sqrt((b ^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) + (3*b*c^2 *d - a*d^3)*e)/b^3))/x) + b*sqrt((b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a ^2*c^2*d^4)*e^2/(a*b^5)) + (3*b*c^2*d - a*d^3)*e)/b^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)*e - (a*b^4*d*x*sqrt(-(b ^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) + (b^3*c^4 - 3*a*b^2* c^2*d^2)*e*x)*sqrt((b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^ 2/(a*b^5)) + (3*b*c^2*d - a*d^3)*e)/b^3))/x) + b*sqrt(-(b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) - (3*b*c^2*d - a*d^3)*e)/b^ 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)*e + (a*b^4*d*x*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) - (b^3*c^4 - 3*a*b^2*c^2*d^2)*e*x)*sqrt(-(b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4* d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) - (3*b*c^2*d - a*d^3)*e)/b^3))/x) - b*sq rt(-(b^3*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)*e^2/(a*b^5)) - (3 *b*c^2*d - a*d^3)*e)/b^3)*log(-((b^2*c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4)*sq rt(d*x + c)*sqrt(e*x)*e - (a*b^4*d*x*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9...
\[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\int \frac {\sqrt {e x} \left (c + d x\right )^{\frac {3}{2}}}{a + b x^{2}}\, dx \] Input:
integrate((e*x)**(1/2)*(d*x+c)**(3/2)/(b*x**2+a),x)
Output:
Integral(sqrt(e*x)*(c + d*x)**(3/2)/(a + b*x**2), x)
\[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}} \sqrt {e x}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^(1/2)*(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^(3/2)*sqrt(e*x)/(b*x^2 + a), x)
Exception generated. \[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^(1/2)*(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\int \frac {\sqrt {e\,x}\,{\left (c+d\,x\right )}^{3/2}}{b\,x^2+a} \,d x \] Input:
int(((e*x)^(1/2)*(c + d*x)^(3/2))/(a + b*x^2),x)
Output:
int(((e*x)^(1/2)*(c + d*x)^(3/2))/(a + b*x^2), x)
\[ \int \frac {\sqrt {e x} (c+d x)^{3/2}}{a+b x^2} \, dx=\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, x}{b \,x^{2}+a}d x \right ) d +\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}}{b \,x^{2}+a}d x \right ) c \right ) \] Input:
int((e*x)^(1/2)*(d*x+c)^(3/2)/(b*x^2+a),x)
Output:
sqrt(e)*(int((sqrt(x)*sqrt(c + d*x)*x)/(a + b*x**2),x)*d + int((sqrt(x)*sq rt(c + d*x))/(a + b*x**2),x)*c)