Integrand size = 37, antiderivative size = 133 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {x \left (d+e x^2\right )^{5/2}}{5 a \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}}+\frac {4 x \left (d+e x^2\right )^{3/2}}{15 a^2 \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {8 x \sqrt {d+e x^2}}{15 a^3 \sqrt {a d+(b d+a e) x^2+b e x^4}} \] Output:
1/5*x*(e*x^2+d)^(5/2)/a/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2)+4/15*x*(e*x^2+d) ^(3/2)/a^2/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)+8/15*x*(e*x^2+d)^(1/2)/a^3/(a *d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\left (d+e x^2\right )^{5/2} \left (15 a^2 x+20 a b x^3+8 b^2 x^5\right )}{15 a^3 \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{5/2}} \] Input:
Integrate[(d + e*x^2)^(7/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2),x]
Output:
((d + e*x^2)^(5/2)*(15*a^2*x + 20*a*b*x^3 + 8*b^2*x^5))/(15*a^3*((a + b*x^ 2)*(d + e*x^2))^(5/2))
Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1395, 209, 209, 208}
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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 1395 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right ) \sqrt {d+e x^2}}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
Input:
Int[(d + e*x^2)^(7/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2),x]
Output:
(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(x/(5*a*(a + b*x^2)^(5/2)) + (4*(x/(3*a*( a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/(5*a)))/Sqrt[a*d + (b* d + a*e)*x^2 + b*e*x^4]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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 = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.47
method | result | size |
default | \(\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, x \left (8 b^{2} x^{4}+20 a b \,x^{2}+15 a^{2}\right )}{15 \sqrt {e \,x^{2}+d}\, \left (b \,x^{2}+a \right )^{3} a^{3}}\) | \(63\) |
orering | \(\frac {x \left (8 b^{2} x^{4}+20 a b \,x^{2}+15 a^{2}\right ) \left (b \,x^{2}+a \right ) \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{15 a^{3} \left (a d +\left (a e +b d \right ) x^{2}+b e \,x^{4}\right )^{\frac {7}{2}}}\) | \(67\) |
gosper | \(\frac {\left (b \,x^{2}+a \right ) x \left (8 b^{2} x^{4}+20 a b \,x^{2}+15 a^{2}\right ) \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{15 a^{3} \left (b e \,x^{4}+a e \,x^{2}+b d \,x^{2}+a d \right )^{\frac {7}{2}}}\) | \(68\) |
Input:
int((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x,method=_RETURNVERB OSE)
Output:
1/15/(e*x^2+d)^(1/2)*((e*x^2+d)*(b*x^2+a))^(1/2)*x*(8*b^2*x^4+20*a*b*x^2+1 5*a^2)/(b*x^2+a)^3/a^3
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {{\left (8 \, b^{2} x^{5} + 20 \, a b x^{3} + 15 \, a^{2} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{15 \, {\left (a^{3} b^{3} e x^{8} + a^{6} d + {\left (a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{6} + 3 \, {\left (a^{4} b^{2} d + a^{5} b e\right )} x^{4} + {\left (3 \, a^{5} b d + a^{6} e\right )} x^{2}\right )}} \] Input:
integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" fricas")
Output:
1/15*(8*b^2*x^5 + 20*a*b*x^3 + 15*a^2*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)/(a^3*b^3*e*x^8 + a^6*d + (a^3*b^3*d + 3*a^4*b^2*e)*x^ 6 + 3*(a^4*b^2*d + a^5*b*e)*x^4 + (3*a^5*b*d + a^6*e)*x^2)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(7/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" maxima")
Output:
integrate((e*x^2 + d)^(7/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm=" giac")
Output:
integrate((e*x^2 + d)^(7/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2), x)
Time = 17.61 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\sqrt {b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d}\,\left (\frac {4\,x^3\,\sqrt {e\,x^2+d}}{3\,a^2\,b^2\,e}+\frac {8\,x^5\,\sqrt {e\,x^2+d}}{15\,a^3\,b\,e}+\frac {x\,\sqrt {e\,x^2+d}}{a\,b^3\,e}\right )}{x^8+\frac {a^3\,d}{b^3\,e}+\frac {x^6\,\left (3\,a\,e+b\,d\right )}{b\,e}+\frac {x^2\,\left (e\,a^6+3\,b\,d\,a^5\right )}{a^3\,b^3\,e}+\frac {3\,a\,x^4\,\left (a\,e+b\,d\right )}{b^2\,e}} \] Input:
int((d + e*x^2)^(7/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2),x)
Output:
((a*d + x^2*(a*e + b*d) + b*e*x^4)^(1/2)*((4*x^3*(d + e*x^2)^(1/2))/(3*a^2 *b^2*e) + (8*x^5*(d + e*x^2)^(1/2))/(15*a^3*b*e) + (x*(d + e*x^2)^(1/2))/( a*b^3*e)))/(x^8 + (a^3*d)/(b^3*e) + (x^6*(3*a*e + b*d))/(b*e) + (x^2*(a^6* e + 3*a^5*b*d))/(a^3*b^3*e) + (3*a*x^4*(a*e + b*d))/(b^2*e))
Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {15 \sqrt {b \,x^{2}+a}\, a^{2} b x +20 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{3} x^{5}-8 \sqrt {b}\, a^{3}-24 \sqrt {b}\, a^{2} b \,x^{2}-24 \sqrt {b}\, a \,b^{2} x^{4}-8 \sqrt {b}\, b^{3} x^{6}}{15 a^{3} b \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right )} \] Input:
int((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)
Output:
(15*sqrt(a + b*x**2)*a**2*b*x + 20*sqrt(a + b*x**2)*a*b**2*x**3 + 8*sqrt(a + b*x**2)*b**3*x**5 - 8*sqrt(b)*a**3 - 24*sqrt(b)*a**2*b*x**2 - 24*sqrt(b )*a*b**2*x**4 - 8*sqrt(b)*b**3*x**6)/(15*a**3*b*(a**3 + 3*a**2*b*x**2 + 3* a*b**2*x**4 + b**3*x**6))