Integrand size = 22, antiderivative size = 95 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a d+b c x}{b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {c^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}} \] Output:
-(b*c*x+a*d)/b/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)-c^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 = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a d-b c x}{b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {2 c^2 \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}} \] Input:
Integrate[x^2/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
(-(a*d) - b*c*x)/(b*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) + (2*c^2*ArcTan[(Sqrt [-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2])])/(-(b*c^2) - a*d^2)^(3/2)
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {601, 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.
\(\displaystyle \int \frac {x^2}{\left (a+b x^2\right )^{3/2} (c+d x)} \, dx\) |
\(\Big \downarrow \) 601 |
\(\displaystyle -\frac {\int -\frac {a c^2}{\left (b c^2+a d^2\right ) (c+d x) \sqrt {b x^2+a}}dx}{a}-\frac {a d+b c x}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a c^2}{\left (b c^2+a d^2\right ) (c+d x) \sqrt {b x^2+a}}dx}{a}-\frac {a d+b c x}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c^2 \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {a d+b c x}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {c^2 \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}-\frac {a d+b c x}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {c^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}-\frac {a d+b c x}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )}\) |
Input:
Int[x^2/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
-((a*d + b*c*x)/(b*(b*c^2 + a*d^2)*Sqrt[a + b*x^2])) - (c^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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs. \(2(87)=174\).
Time = 0.40 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.74
method | result | size |
default | \(-\frac {1}{d b \sqrt {b \,x^{2}+a}}+\frac {c^{2} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}-\frac {c x}{d^{2} \sqrt {b \,x^{2}+a}\, a}\) | \(355\) |
Input:
int(x^2/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/d/b/(b*x^2+a)^(1/2)+c^2/d^3*(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)))-c/d^2/(b*x^2+a)^(1/2)/a *x
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (88) = 176\).
Time = 0.12 (sec) , antiderivative size = 455, normalized size of antiderivative = 4.79 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b^{2} c^{2} x^{2} + a b c^{2}\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (a b c^{2} d + a^{2} d^{3} + {\left (b^{2} c^{3} + a b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4} + {\left (b^{4} c^{4} + 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} x^{2}\right )}}, -\frac {{\left (b^{2} c^{2} x^{2} + a b c^{2}\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + {\left (a b c^{2} d + a^{2} d^{3} + {\left (b^{2} c^{3} + a b c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4} + {\left (b^{4} c^{4} + 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} x^{2}}\right ] \] Input:
integrate(x^2/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((b^2*c^2*x^2 + a*b*c^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*b^3*c^4 + 2*a^2*b^2*c^2*d ^2 + a^3*b*d^4 + (b^4*c^4 + 2*a*b^3*c^2*d^2 + a^2*b^2*d^4)*x^2), -((b^2*c^ 2*x^2 + a*b*c^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*b^3*c^4 + 2*a^2*b^2*c^2*d^2 + a^3*b*d^4 + (b^4*c^4 + 2*a*b^3*c^2*d^2 + a^2*b^2*d^4) *x^2)]
\[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate(x**2/(d*x+c)/(b*x**2+a)**(3/2),x)
Output:
Integral(x**2/((a + b*x**2)**(3/2)*(c + d*x)), x)
Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.80 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {b c^{3} x}{\sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} + \frac {c^{2}}{\sqrt {b x^{2} + a} b c^{2} d + \sqrt {b x^{2} + a} a d^{3}} - \frac {c x}{\sqrt {b x^{2} + a} a d^{2}} + \frac {c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} - \frac {1}{\sqrt {b x^{2} + a} b d} \] Input:
integrate(x^2/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
b*c^3*x/(sqrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) + c^2/(sqr t(b*x^2 + a)*b*c^2*d + sqrt(b*x^2 + a)*a*d^3) - c*x/(sqrt(b*x^2 + a)*a*d^2 ) + c^2*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^3) - 1/(sqrt(b*x^2 + a)*b*d)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (88) = 176\).
Time = 0.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.91 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 \, c^{2} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {\frac {{\left (b^{2} c^{3} + a b c d^{2}\right )} x}{b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2} + a^{2} b d^{4}} + \frac {a b c^{2} d + a^{2} d^{3}}{b^{3} c^{4} + 2 \, a b^{2} c^{2} d^{2} + a^{2} b d^{4}}}{\sqrt {b x^{2} + a}} \] Input:
integrate(x^2/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
-2*c^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/ (b^3*c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4) + (a*b*c^2*d + a^2*d^3)/(b^3*c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4))/sqrt(b*x^2 + a)
Timed out. \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int(x^2/((a + b*x^2)^(3/2)*(c + d*x)),x)
Output:
int(x^2/((a + b*x^2)^(3/2)*(c + d*x)), x)
Time = 0.63 (sec) , antiderivative size = 2355, normalized size of antiderivative = 24.79 \[ \int \frac {x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^2/(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*b*c**3 - 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))*b**2*c**3*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))*a**2*b*c**2 *d**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**2*c**4 - 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*b**2*c**2*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**3*c**4*x**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)*log( - sqrt(2*sqrt(b)*sq rt(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*b*c**3 - 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)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c...