Integrand size = 37, antiderivative size = 104 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {(b d-a e) x \sqrt {d+e x^2}}{a b \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{b^{3/2}} \] Output:
(-a*e+b*d)*x*(e*x^2+d)^(1/2)/a/b/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+e*arcta nh(b^(1/2)*x*(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2))/b^(3/2)
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} (b d-a e) x-a e \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{a b^{3/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:
Integrate[(d + e*x^2)^(5/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
Output:
(Sqrt[d + e*x^2]*(Sqrt[b]*(b*d - a*e)*x - a*e*Sqrt[a + b*x^2]*Log[-(Sqrt[b ]*x) + Sqrt[a + b*x^2]]))/(a*b^(3/2)*Sqrt[(a + b*x^2)*(d + e*x^2)])
Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1395, 298, 224, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1395 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {e x^2+d}{\left (b x^2+a\right )^{3/2}}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \int \frac {1}{\sqrt {b x^2+a}}dx}{b}+\frac {x (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b}+\frac {x (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}+\frac {x (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
Input:
Int[(d + e*x^2)^(5/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
Output:
(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(((b*d - a*e)*x)/(a*b*Sqrt[a + b*x^2]) + (e*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)))/Sqrt[a*d + (b*d + a*e)* x^2 + b*e*x^4]
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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a e \sqrt {b \,x^{2}+a}-a e x \sqrt {b}+b^{\frac {3}{2}} d x \right )}{b^{\frac {3}{2}} \sqrt {e \,x^{2}+d}\, \left (b \,x^{2}+a \right ) a}\) | \(86\) |
Input:
int((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x,method=_RETURNVERB OSE)
Output:
((e*x^2+d)*(b*x^2+a))^(1/2)/b^(3/2)*(ln(b^(1/2)*x+(b*x^2+a)^(1/2))*a*e*(b* x^2+a)^(1/2)-a*e*x*b^(1/2)+b^(3/2)*d*x)/(e*x^2+d)^(1/2)/(b*x^2+a)/a
Time = 0.09 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.50 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\left [\frac {2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (b^{2} d - a b e\right )} \sqrt {e x^{2} + d} x + {\left (a b e^{2} x^{4} + a^{2} d e + {\left (a b d e + a^{2} e^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b e x^{4} + {\left (2 \, b d + a e\right )} x^{2} + 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} \sqrt {b} x + a d}{e x^{2} + d}\right )}{2 \, {\left (a b^{3} e x^{4} + a^{2} b^{2} d + {\left (a b^{3} d + a^{2} b^{2} e\right )} x^{2}\right )}}, \frac {\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (b^{2} d - a b e\right )} \sqrt {e x^{2} + d} x - {\left (a b e^{2} x^{4} + a^{2} d e + {\left (a b d e + a^{2} e^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-b} x}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d}}\right )}{a b^{3} e x^{4} + a^{2} b^{2} d + {\left (a b^{3} d + a^{2} b^{2} e\right )} x^{2}}\right ] \] Input:
integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" fricas")
Output:
[1/2*(2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(b^2*d - a*b*e)*sqrt(e*x^2 + d)*x + (a*b*e^2*x^4 + a^2*d*e + (a*b*d*e + a^2*e^2)*x^2)*sqrt(b)*log((2*b *e*x^4 + (2*b*d + a*e)*x^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt( e*x^2 + d)*sqrt(b)*x + a*d)/(e*x^2 + d)))/(a*b^3*e*x^4 + a^2*b^2*d + (a*b^ 3*d + a^2*b^2*e)*x^2), (sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(b^2*d - a*b *e)*sqrt(e*x^2 + d)*x - (a*b*e^2*x^4 + a^2*d*e + (a*b*d*e + a^2*e^2)*x^2)* sqrt(-b)*arctan(sqrt(e*x^2 + d)*sqrt(-b)*x/sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)))/(a*b^3*e*x^4 + a^2*b^2*d + (a*b^3*d + a^2*b^2*e)*x^2)]
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {5}{2}}}{\left (\left (a + b x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**(5/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(3/2),x)
Output:
Integral((d + e*x**2)**(5/2)/((a + b*x**2)*(d + e*x**2))**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" maxima")
Output:
integrate((e*x^2 + d)^(5/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(3/2), x)
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.48 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=-\frac {e \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} + \frac {{\left (b d - a e\right )} x}{\sqrt {b x^{2} + a} a b} \] Input:
integrate((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" giac")
Output:
-e*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + (b*d - a*e)*x/(sqrt(b* x^2 + a)*a*b)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)^(5/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2),x)
Output:
int((d + e*x^2)^(5/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2), x)
Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a b e x +\sqrt {b \,x^{2}+a}\, b^{2} d x +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} e +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b e \,x^{2}-\sqrt {b}\, a^{2} e +\sqrt {b}\, a b d -\sqrt {b}\, a b e \,x^{2}+\sqrt {b}\, b^{2} d \,x^{2}}{a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:
int((e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x)
Output:
( - sqrt(a + b*x**2)*a*b*e*x + sqrt(a + b*x**2)*b**2*d*x + sqrt(b)*log((sq rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*e + sqrt(b)*log((sqrt(a + b*x**2 ) + sqrt(b)*x)/sqrt(a))*a*b*e*x**2 - sqrt(b)*a**2*e + sqrt(b)*a*b*d - sqrt (b)*a*b*e*x**2 + sqrt(b)*b**2*d*x**2)/(a*b**2*(a + b*x**2))