Integrand size = 28, antiderivative size = 152 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d) x}{3 a (b e-a f) \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^2 c e+2 a^2 d f+a b (d e-5 c f)\right ) x}{3 a^2 (b e-a f)^2 \sqrt {a+b x^2}}-\frac {f (d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{5/2}} \] Output:
1/3*(-a*d+b*c)*x/a/(-a*f+b*e)/(b*x^2+a)^(3/2)+1/3*(2*b^2*c*e+2*a^2*d*f+a*b *(-5*c*f+d*e))*x/a^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2)-f*(-c*f+d*e)*arctanh((-a *f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/(-a*f+b*e)^(5/2)
Time = 0.77 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {x \left (3 a^3 d f+2 b^3 c e x^2+2 a^2 b f \left (-3 c+d x^2\right )+a b^2 \left (3 c e+d e x^2-5 c f x^2\right )\right )}{3 a^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2}}+\frac {f (d e-c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (-b e+a f)^{5/2}} \] Input:
Integrate[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(x*(3*a^3*d*f + 2*b^3*c*e*x^2 + 2*a^2*b*f*(-3*c + d*x^2) + a*b^2*(3*c*e + d*e*x^2 - 5*c*f*x^2)))/(3*a^2*(b*e - a*f)^2*(a + b*x^2)^(3/2)) + (f*(d*e - c*f)*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[ -(b*e) + a*f])])/(Sqrt[e]*(-(b*e) + a*f)^(5/2))
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {402, 25, 402, 27, 291, 221}
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.
\(\displaystyle \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}-\frac {\int -\frac {2 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}dx}{3 a (b e-a f)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 (b c-a d) f x^2+2 b c e+a d e-3 a c f}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )}dx}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {\int \frac {3 a^2 f (d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{b e-a f}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b e-a f}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {x \left (2 a^2 d f+a b (d e-5 c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} (b e-a f)}-\frac {3 a f (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{3/2}}}{3 a (b e-a f)}+\frac {x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} (b e-a f)}\) |
Input:
Int[(c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
Output:
((b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^2)^(3/2)) + (((2*b^2*c*e + 2*a^2 *d*f + a*b*(d*e - 5*c*f))*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]) - (3*a*f*(d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*( b*e - a*f)^(3/2)))/(3*a*(b*e - a*f))
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 0.94 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\frac {-a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (c f -d e \right ) f \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (a^{3} d f -2 \left (-\frac {x^{2} d}{3}+c \right ) b f \,a^{2}+\left (e \left (\frac {x^{2} d}{3}+c \right )-\frac {5 c f \,x^{2}}{3}\right ) b^{2} a +\frac {2 b^{3} c e \,x^{2}}{3}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{\sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (a f -b e \right )^{2} a^{2}}\) | \(156\) |
default | \(\text {Expression too large to display}\) | \(1436\) |
Input:
int((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(-a^2*(b*x^2+a)^(3/2)*(c*f-d*e)*f*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e) ^(1/2))+(a^3*d*f-2*(-1/3*x^2*d+c)*b*f*a^2+(e*(1/3*x^2*d+c)-5/3*c*f*x^2)*b^ 2*a+2/3*b^3*c*e*x^2)*((a*f-b*e)*e)^(1/2)*x)/((a*f-b*e)*e)^(1/2)/(b*x^2+a)^ (3/2)/(a*f-b*e)^2/a^2
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (134) = 268\).
Time = 2.80 (sec) , antiderivative size = 940, normalized size of antiderivative = 6.18 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="fricas")
Output:
[-1/12*(3*(a^4*d*e*f - a^4*c*f^2 + (a^2*b^2*d*e*f - a^2*b^2*c*f^2)*x^4 + 2 *(a^3*b*d*e*f - a^3*b*c*f^2)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8* a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b *e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e *f*x^2 + e^2)) - 4*(((2*b^4*c + a*b^3*d)*e^3 - (7*a*b^3*c - a^2*b^2*d)*e^2 *f + (5*a^2*b^2*c - 2*a^3*b*d)*e*f^2)*x^3 + 3*(a*b^3*c*e^3 - (3*a^2*b^2*c - a^3*b*d)*e^2*f + (2*a^3*b*c - a^4*d)*e*f^2)*x)*sqrt(b*x^2 + a))/(a^4*b^3 *e^4 - 3*a^5*b^2*e^3*f + 3*a^6*b*e^2*f^2 - a^7*e*f^3 + (a^2*b^5*e^4 - 3*a^ 3*b^4*e^3*f + 3*a^4*b^3*e^2*f^2 - a^5*b^2*e*f^3)*x^4 + 2*(a^3*b^4*e^4 - 3* a^4*b^3*e^3*f + 3*a^5*b^2*e^2*f^2 - a^6*b*e*f^3)*x^2), 1/6*(3*(a^4*d*e*f - a^4*c*f^2 + (a^2*b^2*d*e*f - a^2*b^2*c*f^2)*x^4 + 2*(a^3*b*d*e*f - a^3*b* c*f^2)*x^2)*sqrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 + a)/((b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2 *e*f)*x)) + 2*(((2*b^4*c + a*b^3*d)*e^3 - (7*a*b^3*c - a^2*b^2*d)*e^2*f + (5*a^2*b^2*c - 2*a^3*b*d)*e*f^2)*x^3 + 3*(a*b^3*c*e^3 - (3*a^2*b^2*c - a^3 *b*d)*e^2*f + (2*a^3*b*c - a^4*d)*e*f^2)*x)*sqrt(b*x^2 + a))/(a^4*b^3*e^4 - 3*a^5*b^2*e^3*f + 3*a^6*b*e^2*f^2 - a^7*e*f^3 + (a^2*b^5*e^4 - 3*a^3*b^4 *e^3*f + 3*a^4*b^3*e^2*f^2 - a^5*b^2*e*f^3)*x^4 + 2*(a^3*b^4*e^4 - 3*a^4*b ^3*e^3*f + 3*a^5*b^2*e^2*f^2 - a^6*b*e*f^3)*x^2)]
Timed out. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)/(b*x**2+a)**(5/2)/(f*x**2+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (134) = 268\).
Time = 0.14 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.34 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {{\left (\frac {{\left (2 \, b^{6} c e^{3} + a b^{5} d e^{3} - 9 \, a b^{5} c e^{2} f + 12 \, a^{2} b^{4} c e f^{2} - 3 \, a^{3} b^{3} d e f^{2} - 5 \, a^{3} b^{3} c f^{3} + 2 \, a^{4} b^{2} d f^{3}\right )} x^{2}}{a^{2} b^{5} e^{4} - 4 \, a^{3} b^{4} e^{3} f + 6 \, a^{4} b^{3} e^{2} f^{2} - 4 \, a^{5} b^{2} e f^{3} + a^{6} b f^{4}} + \frac {3 \, {\left (a b^{5} c e^{3} - 4 \, a^{2} b^{4} c e^{2} f + a^{3} b^{3} d e^{2} f + 5 \, a^{3} b^{3} c e f^{2} - 2 \, a^{4} b^{2} d e f^{2} - 2 \, a^{4} b^{2} c f^{3} + a^{5} b d f^{3}\right )}}{a^{2} b^{5} e^{4} - 4 \, a^{3} b^{4} e^{3} f + 6 \, a^{4} b^{3} e^{2} f^{2} - 4 \, a^{5} b^{2} e f^{3} + a^{6} b f^{4}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {{\left (b d^{2} e^{2} f - 2 \, b c d e f^{2} + b c^{2} f^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} d e f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} c f^{2} + 2 \, b^{\frac {3}{2}} d e^{2} - 2 \, b^{\frac {3}{2}} c e f - a \sqrt {b} d e f + a \sqrt {b} c f^{2}}{2 \, {\left (\sqrt {-b e^{2} + a e f} b d e - \sqrt {-b e^{2} + a e f} b c f\right )}}\right )}{{\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} {\left (b d e - b c f\right )} \sqrt {-b e^{2} + a e f}} \] Input:
integrate((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="giac")
Output:
1/3*((2*b^6*c*e^3 + a*b^5*d*e^3 - 9*a*b^5*c*e^2*f + 12*a^2*b^4*c*e*f^2 - 3 *a^3*b^3*d*e*f^2 - 5*a^3*b^3*c*f^3 + 2*a^4*b^2*d*f^3)*x^2/(a^2*b^5*e^4 - 4 *a^3*b^4*e^3*f + 6*a^4*b^3*e^2*f^2 - 4*a^5*b^2*e*f^3 + a^6*b*f^4) + 3*(a*b ^5*c*e^3 - 4*a^2*b^4*c*e^2*f + a^3*b^3*d*e^2*f + 5*a^3*b^3*c*e*f^2 - 2*a^4 *b^2*d*e*f^2 - 2*a^4*b^2*c*f^3 + a^5*b*d*f^3)/(a^2*b^5*e^4 - 4*a^3*b^4*e^3 *f + 6*a^4*b^3*e^2*f^2 - 4*a^5*b^2*e*f^3 + a^6*b*f^4))*x/(b*x^2 + a)^(3/2) + (b*d^2*e^2*f - 2*b*c*d*e*f^2 + b*c^2*f^3)*arctan(1/2*((sqrt(b)*x - sqrt (b*x^2 + a))^2*sqrt(b)*d*e*f - (sqrt(b)*x - sqrt(b*x^2 + a))^2*sqrt(b)*c*f ^2 + 2*b^(3/2)*d*e^2 - 2*b^(3/2)*c*e*f - a*sqrt(b)*d*e*f + a*sqrt(b)*c*f^2 )/(sqrt(-b*e^2 + a*e*f)*b*d*e - sqrt(-b*e^2 + a*e*f)*b*c*f))/((b^2*e^2 - 2 *a*b*e*f + a^2*f^2)*(b*d*e - b*c*f)*sqrt(-b*e^2 + a*e*f))
Timed out. \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {d\,x^2+c}{{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)),x)
Output:
int((c + d*x^2)/((a + b*x^2)^(5/2)*(e + f*x^2)), x)
Time = 0.21 (sec) , antiderivative size = 1425, normalized size of antiderivative = 9.38 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)/(b*x^2+a)^(5/2)/(f*x^2+e),x)
Output:
( - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x **2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b*c*f**2 + 3*sqrt(e)*sqr t(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sq rt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b*d*e*f - 6*sqrt(e)*sqrt(a*f - b*e)*atan( (sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)* sqrt(b)))*a**3*b**2*c*f**2*x**2 + 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b))) *a**3*b**2*d*e*f*x**2 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3 *c*f**2*x**4 + 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*s qrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*d*e*f*x* *4 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b* x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b*c*f**2 + 3*sqrt(e)*sq rt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*s qrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b*d*e*f - 6*sqrt(e)*sqrt(a*f - b*e)*atan ((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e) *sqrt(b)))*a**3*b**2*c*f**2*x**2 + 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a* f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)) )*a**3*b**2*d*e*f*x**2 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*...