Integrand size = 22, antiderivative size = 432 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\sqrt [4]{b} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}} \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2}}-\frac {\sqrt [4]{b} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}} \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {\sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} a \sqrt {b c^2+a d^2}} \] Output:
1/2*b^(1/4)*(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)*arctan(((b^(1/2)*c+(a*d ^2+b*c^2)^(1/2))^(1/2)-2^(1/2)*b^(1/4)*(d*x+c)^(1/2))/(-b^(1/2)*c+(a*d^2+b *c^2)^(1/2))^(1/2))*2^(1/2)/a/(a*d^2+b*c^2)^(1/2)-1/2*b^(1/4)*(-b^(1/2)*c+ (a*d^2+b*c^2)^(1/2))^(1/2)*arctan(((b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+2 ^(1/2)*b^(1/4)*(d*x+c)^(1/2))/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2))*2^(1 /2)/a/(a*d^2+b*c^2)^(1/2)-2*arctanh((d*x+c)^(1/2)/c^(1/2))/a/c^(1/2)+1/2*b ^(1/4)*(b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)*arctanh(2^(1/2)*b^(1/4)*(b^(1 /2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)*(d*x+c)^(1/2)/((a*d^2+b*c^2)^(1/2)+b^(1/2 )*(d*x+c)))*2^(1/2)/a/(a*d^2+b*c^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.46 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\sqrt {-b c+i \sqrt {a} \sqrt {b} d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{a} \] Input:
Integrate[1/(x*Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
((Sqrt[b]*ArcTan[(Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[ b]*c + I*Sqrt[a]*d)])/Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d] + (Sqrt[b]*ArcTan [(Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - I*Sqrt[a] *d)])/Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d] - (2*ArcTanh[Sqrt[c + d*x]/Sqrt[c ]])/Sqrt[c])/a
Time = 1.59 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {561, 25, 27, 1484, 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.
| \(\displaystyle \int \frac {1}{x \left (a+b x^2\right ) \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {2 \int \frac {1}{x \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \int -\frac {1}{x \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int -\frac {1}{d x \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -2 \int \left (\frac {b d x}{a \left (b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2\right )}-\frac {1}{a d x}\right )d\sqrt {c+d x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (-\frac {\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}} \text {arctanh}\left (\frac {\sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}\right )}{2 \sqrt {2} a \sqrt {a d^2+b c^2}}+\frac {\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}} \text {arctanh}\left (\frac {\sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}\right )}{2 \sqrt {2} a \sqrt {a d^2+b c^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {\sqrt [4]{b} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {a d^2+b c^2}+\sqrt {b} (c+d x)\right )}{4 \sqrt {2} a \sqrt {a d^2+b c^2}}-\frac {\sqrt [4]{b} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt {c+d x} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {a d^2+b c^2}+\sqrt {b} (c+d x)\right )}{4 \sqrt {2} a \sqrt {a d^2+b c^2}}\right )\) |
Input:
Int[1/(x*Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
-2*(ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a*Sqrt[c]) - (b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] - Sqr t[2]*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqr t[2]*a*Sqrt[b*c^2 + a*d^2]) + (b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2 ]]*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] + Sqrt[2]*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[2]*a*Sqrt[b*c^2 + a*d^2]) + (b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Log[Sqrt[b*c^2 + a*d^2] - Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d *x] + Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*a*Sqrt[b*c^2 + a*d^2]) - (b^(1/4)*Sqr t[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Log[Sqrt[b*c^2 + a*d^2] + Sqrt[2]*b^(1/ 4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x) ])/(4*Sqrt[2]*a*Sqrt[b*c^2 + a*d^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(340)=680\).
Time = 0.73 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.69
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (\left (\sqrt {c}\, \sqrt {a \,d^{2}+b \,c^{2}}+\sqrt {b}\, c^{\frac {3}{2}}\right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, b -c^{\frac {5}{2}} b^{\frac {3}{2}}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (\left (\sqrt {c}\, \sqrt {a \,d^{2}+b \,c^{2}}+\sqrt {b}\, c^{\frac {3}{2}}\right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-c^{\frac {3}{2}} \sqrt {a \,d^{2}+b \,c^{2}}\, b -c^{\frac {5}{2}} b^{\frac {3}{2}}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}+d^{2} \left (2 \sqrt {a \,d^{2}+b \,c^{2}}\, \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {b}+\left (\sqrt {c}\, \sqrt {a \,d^{2}+b \,c^{2}}\, b -c^{\frac {3}{2}} b^{\frac {3}{2}}\right ) \left (\arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )-\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )\right )\right ) a}{\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {c}\, d^{2} a^{2}}\) | \(729\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2084\) |
| default | \(\text {Expression too large to display}\) | \(2084\) |
Input:
int(1/x/(d*x+c)^(1/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-(1/4*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/ 2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*((c^(1/2)*(a*d^2+b*c^2)^(1/2)+b^ (1/2)*c^(3/2))*((a*d^2+b*c^2)*b)^(1/2)-c^(3/2)*(a*d^2+b*c^2)^(1/2)*b-c^(5/ 2)*b^(3/2))*ln(b^(1/2)*(d*x+c)-(d*x+c)^(1/2)*(2*((a*d^2+b*c^2)*b)^(1/2)+2* b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))-1/4*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2 )*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*(( c^(1/2)*(a*d^2+b*c^2)^(1/2)+b^(1/2)*c^(3/2))*((a*d^2+b*c^2)*b)^(1/2)-c^(3/ 2)*(a*d^2+b*c^2)^(1/2)*b-c^(5/2)*b^(3/2))*ln(b^(1/2)*(d*x+c)+(d*x+c)^(1/2) *(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))+d^2*(2*(a*d^ 2+b*c^2)^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*(4*(a*d^2+b*c^2)^(1/2)*b^(1/ 2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*b^(1/2)+(c^(1/2)*(a*d^2+b*c^2)^( 1/2)*b-c^(3/2)*b^(3/2))*(arctan((2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2) *b)^(1/2)+2*b*c)^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b) ^(1/2)-2*b*c)^(1/2))-arctan((-2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b) ^(1/2)+2*b*c)^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1 /2)-2*b*c)^(1/2))))*a)/b^(1/2)/(a*d^2+b*c^2)^(1/2)/(4*(a*d^2+b*c^2)^(1/2)* b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)/c^(1/2)/d^2/a^2
Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (342) = 684\).
Time = 0.15 (sec) , antiderivative size = 2017, normalized size of antiderivative = 4.67 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="fricas")
Output:
[1/2*(a*c*sqrt((b*c + (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a ^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))*log(sqrt(d*x + c)*b + (a* b*c - (a^3*b*c^2 + a^4*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a ^5*d^4)))*sqrt((b*c + (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a ^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))) - a*c*sqrt((b*c + (a^2*b *c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a ^2*b*c^2 + a^3*d^2))*log(sqrt(d*x + c)*b - (a*b*c - (a^3*b*c^2 + a^4*d^2)* sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))*sqrt((b*c + (a^2*b *c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a ^2*b*c^2 + a^3*d^2))) + a*c*sqrt((b*c - (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/ (a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))*log(sqr t(d*x + c)*b + (a*b*c + (a^3*b*c^2 + a^4*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2 *a^4*b*c^2*d^2 + a^5*d^4)))*sqrt((b*c - (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/ (a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))) - a*c* sqrt((b*c - (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d ^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))*log(sqrt(d*x + c)*b - (a*b*c + (a^3 *b*c^2 + a^4*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)))* sqrt((b*c - (a^2*b*c^2 + a^3*d^2)*sqrt(-b*d^2/(a^3*b^2*c^4 + 2*a^4*b*c^2*d ^2 + a^5*d^4)))/(a^2*b*c^2 + a^3*d^2))) + 2*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x))/(a*c), 1/2*(a*c*sqrt((b*c + (a^2*b*c^2 + a^3*d^...
\[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int \frac {1}{x \left (a + b x^{2}\right ) \sqrt {c + d x}}\, dx \] Input:
integrate(1/x/(d*x+c)**(1/2)/(b*x**2+a),x)
Output:
Integral(1/(x*(a + b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x + c} x} \,d x } \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)*sqrt(d*x + c)*x), x)
Time = 0.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {{\left ({\left | a \right |} {\left | b \right |} {\left | d \right |} + \sqrt {-a b} c {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} + a^{2} d^{2}\right )} a b}}{a b}}}\right )}{\sqrt {-b^{2} c - \sqrt {-a b} b d} {\left (a^{2} d + \sqrt {-a b} a c\right )}} - \frac {{\left ({\left | a \right |} {\left | b \right |} {\left | d \right |} - \sqrt {-a b} c {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} + a^{2} d^{2}\right )} a b}}{a b}}}\right )}{\sqrt {-b^{2} c + \sqrt {-a b} b d} {\left (a^{2} d - \sqrt {-a b} a c\right )}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="giac")
Output:
-(abs(a)*abs(b)*abs(d) + sqrt(-a*b)*c*abs(b))*arctan(sqrt(d*x + c)/sqrt(-( a*b*c + sqrt(a^2*b^2*c^2 - (a*b*c^2 + a^2*d^2)*a*b))/(a*b)))/(sqrt(-b^2*c - sqrt(-a*b)*b*d)*(a^2*d + sqrt(-a*b)*a*c)) - (abs(a)*abs(b)*abs(d) - sqrt (-a*b)*c*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b*c - sqrt(a^2*b^2*c^2 - (a *b*c^2 + a^2*d^2)*a*b))/(a*b)))/(sqrt(-b^2*c + sqrt(-a*b)*b*d)*(a^2*d - sq rt(-a*b)*a*c)) + 2*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c))
Time = 8.65 (sec) , antiderivative size = 2735, normalized size of antiderivative = 6.33 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x*(a + b*x^2)*(c + d*x)^(1/2)),x)
Output:
atan((((((512*a^4*b^4*d^10 + (512*a^5*b^4*d^10 + 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1/2)*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) + 384*a^3*b^5*c^2*d^8)*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b* c^2)))^(1/2) - 576*a^2*b^5*c*d^8*(c + d*x)^(1/2))*((d*(-a^5*b)^(1/2) + a^2 *b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) - 96*a*b^5*c*d^8)*((d*(-a^5*b)^(1/2 ) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) + 96*b^5*d^8*(c + d*x)^(1/2) )*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2)*1i - (((( 512*a^4*b^4*d^10 - (512*a^5*b^4*d^10 + 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1/2 )*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) + 384*a^3 *b^5*c^2*d^8)*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/ 2) + 576*a^2*b^5*c*d^8*(c + d*x)^(1/2))*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*( a^5*d^2 + a^4*b*c^2)))^(1/2) - 96*a*b^5*c*d^8)*((d*(-a^5*b)^(1/2) + a^2*b* c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) - 96*b^5*d^8*(c + d*x)^(1/2))*((d*(-a^ 5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2)*1i)/(((((512*a^4*b^ 4*d^10 + (512*a^5*b^4*d^10 + 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1/2)*((d*(-a^ 5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) + 384*a^3*b^5*c^2*d ^8)*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) - 576*a ^2*b^5*c*d^8*(c + d*x)^(1/2))*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 + a^4*b*c^2)))^(1/2) - 96*a*b^5*c*d^8)*((d*(-a^5*b)^(1/2) + a^2*b*c)/(4*(a^5 *d^2 + a^4*b*c^2)))^(1/2) + 96*b^5*d^8*(c + d*x)^(1/2))*((d*(-a^5*b)^(1...
Time = 0.21 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {2 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}-b c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}-2 \sqrt {b}\, \sqrt {d x +c}}{\sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}-b c}\, \sqrt {2}}\right ) c -2 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}-b c}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}+2 \sqrt {b}\, \sqrt {d x +c}}{\sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}-b c}\, \sqrt {2}}\right ) c -\sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {d x +c}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}+\sqrt {a \,d^{2}+b \,c^{2}}+\sqrt {b}\, c +\sqrt {b}\, d x \right ) c +\sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {d x +c}\, \sqrt {\sqrt {b}\, \sqrt {a \,d^{2}+b \,c^{2}}+b c}\, \sqrt {2}+\sqrt {a \,d^{2}+b \,c^{2}}+\sqrt {b}\, c +\sqrt {b}\, d x \right ) c +4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2}+4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b \,c^{2}-4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2}-4 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b \,c^{2}}{4 a c \left (a \,d^{2}+b \,c^{2}\right )} \] Input:
int(1/x/(d*x+c)^(1/2)/(b*x^2+a),x)
Output:
(2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2) *atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt( c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*c - 2*sqrt( a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((s qrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + 2*sqrt(b)*sqrt(c + d*x) )/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*c - sqrt(a*d**2 + b *c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2)*log( - sqrt(c + d *x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + sqrt(a*d**2 + b*c* *2) + sqrt(b)*c + sqrt(b)*d*x)*c + sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt (a*d**2 + b*c**2) + b*c)*sqrt(2)*log(sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d** 2 + b*c**2) + b*c)*sqrt(2) + sqrt(a*d**2 + b*c**2) + sqrt(b)*c + sqrt(b)*d *x)*c + 4*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*d**2 + 4*sqrt(c)*log(sqrt (c + d*x) - sqrt(c))*b*c**2 - 4*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*d** 2 - 4*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*b*c**2)/(4*a*c*(a*d**2 + b*c**2 ))