Integrand size = 19, antiderivative size = 94 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {a d+b c x}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {d^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}} \] Output:
(b*c*x+a*d)/a/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)-d^2*arctanh((-b*c*x+a*d)/(a*d^ 2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2)^(3/2)
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {a d+b c x}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {2 d^2 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}} \] Input:
Integrate[1/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
(a*d + b*c*x)/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) + (2*d^2*ArcTan[(Sqrt[b] *(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2) - a*d^2 )^(3/2)
Time = 0.37 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {496, 25, 27, 488, 219}
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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} (c+d x)} \, dx\) |
\(\Big \downarrow \) 496 |
\(\displaystyle \frac {a d+b c x}{a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}-\frac {\int -\frac {a d^2}{(c+d x) \sqrt {b x^2+a}}dx}{a \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a d^2}{(c+d x) \sqrt {b x^2+a}}dx}{a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d^2 \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}+\frac {a d+b c x}{a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {a d+b c x}{a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}-\frac {d^2 \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a d+b c x}{a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}-\frac {d^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}\) |
Input:
Int[1/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
(a*d + b*c*x)/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) - (d^2*ArcTanh[(a*d - b* c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^(3/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(86)=172\).
Time = 0.38 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.35
method | result | size |
default | \(\frac {\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{d}\) | \(315\) |
Input:
int(1/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4* b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^ 2+b*c^2)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+ c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2 )/d^2)^(1/2))/(x+c/d)))
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (87) = 174\).
Time = 0.13 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.85 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (a b d^{2} x^{2} + a^{2} d^{2}\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (a b c^{2} d + a^{2} d^{3} + {\left (b^{2} c^{3} + a b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4} + {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} x^{2}\right )}}, -\frac {{\left (a b d^{2} x^{2} + a^{2} d^{2}\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (a b c^{2} d + a^{2} d^{3} + {\left (b^{2} c^{3} + a b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{2} c^{4} + 2 \, a^{3} b c^{2} d^{2} + a^{4} d^{4} + {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}\right )} x^{2}}\right ] \] Input:
integrate(1/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((a*b*d^2*x^2 + a^2*d^2)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c*d*x - a*b*c ^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(a*b*c^2*d + a^2*d^ 3 + (b^2*c^3 + a*b*c*d^2)*x)*sqrt(b*x^2 + a))/(a^2*b^2*c^4 + 2*a^3*b*c^2*d ^2 + a^4*d^4 + (a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)*x^2), -((a*b*d^ 2*x^2 + a^2*d^2)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) - (a* b*c^2*d + a^2*d^3 + (b^2*c^3 + a*b*c*d^2)*x)*sqrt(b*x^2 + a))/(a^2*b^2*c^4 + 2*a^3*b*c^2*d^2 + a^4*d^4 + (a*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4) *x^2)]
\[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate(1/(d*x+c)/(b*x**2+a)**(3/2),x)
Output:
Integral(1/((a + b*x**2)**(3/2)*(c + d*x)), x)
Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {b c x}{\sqrt {b x^{2} + a} a b c^{2} + \sqrt {b x^{2} + a} a^{2} d^{2}} + \frac {1}{\frac {\sqrt {b x^{2} + a} b c^{2}}{d} + \sqrt {b x^{2} + a} a d} + \frac {\operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d} \] Input:
integrate(1/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
b*c*x/(sqrt(b*x^2 + a)*a*b*c^2 + sqrt(b*x^2 + a)*a^2*d^2) + 1/(sqrt(b*x^2 + a)*b*c^2/d + sqrt(b*x^2 + a)*a*d) + arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c )) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (87) = 174\).
Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 \, d^{2} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {\frac {{\left (b^{2} c^{3} + a b c d^{2}\right )} x}{a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}} + \frac {a b c^{2} d + a^{2} d^{3}}{a b^{2} c^{4} + 2 \, a^{2} b c^{2} d^{2} + a^{3} d^{4}}}{\sqrt {b x^{2} + a}} \] Input:
integrate(1/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
-2*d^2*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2 + a*d^2)*sqrt(-b*c^2 - a*d^2)) + ((b^2*c^3 + a*b*c*d^2)*x/ (a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4) + (a*b*c^2*d + a^2*d^3)/(a*b^2*c^4 + 2*a^2*b*c^2*d^2 + a^3*d^4))/sqrt(b*x^2 + a)
Timed out. \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x)),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x)), x)
Time = 0.68 (sec) , antiderivative size = 2247, normalized size of antiderivative = 23.90 \[ \int \frac {1}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/(d*x+c)/(b*x^2+a)^(3/2),x)
Output:
( - 2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)* sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt( b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*c - 2*sqrt(b)*sqrt(2*sq rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)*a tan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2 )*c - a*d**2 - 2*b*c**2))*b*c*x**2 - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2 )*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sq rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*d**2 - 2*sqrt(2*s qrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2) *d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c* *2))*a*b*c**2 - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c* *2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b *c**2)*c - a*d**2 - 2*b*c**2))*a*b*d**2*x**2 - 2*sqrt(2*sqrt(b)*sqrt(a*d** 2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x) /sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*b**2*c**2*x* *2 - sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2)*s qrt(a*d**2 + b*c**2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d** 2 + 2*b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d*x)*a*c - sqrt(b)*sqrt(2*sqr t(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2)*sqrt(a*d**2 + b*c**2)*lo g( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqrt...