Integrand size = 48, antiderivative size = 194 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 g \sqrt {d+e x}}{3 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \]
-2/3*(e*x+d)^(3/2)/(-a*e*g+c*d*f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/ (g*x+f)^(1/2)+8/3*g*(e*x+d)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^(1/2)/(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+16/3*g^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ (1/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(1/2)/(g*x+f)^(1/2)
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.53 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (3 a^2 e^2 g^2+6 a c d e g (f+2 g x)+c^2 d^2 \left (-f^2+4 f g x+8 g^2 x^2\right )\right )}{3 (c d f-a e g)^3 ((a e+c d x) (d+e x))^{3/2} \sqrt {f+g x}} \]
(2*(d + e*x)^(3/2)*(3*a^2*e^2*g^2 + 6*a*c*d*e*g*(f + 2*g*x) + c^2*d^2*(-f^ 2 + 4*f*g*x + 8*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*((a*e + c*d*x)*(d + e*x))^ (3/2)*Sqrt[f + g*x])
Time = 0.44 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1252, 1252, 1248}
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.
\(\displaystyle \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1252 |
\(\displaystyle -\frac {4 g \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 (c d f-a e g)}-\frac {2 (d+e x)^{3/2}}{3 \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\) |
\(\Big \downarrow \) 1252 |
\(\displaystyle -\frac {4 g \left (-\frac {2 g \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\right )}{3 (c d f-a e g)}-\frac {2 (d+e x)^{3/2}}{3 \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\) |
\(\Big \downarrow \) 1248 |
\(\displaystyle -\frac {2 (d+e x)^{3/2}}{3 \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}-\frac {4 g \left (-\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\right )}{3 (c d f-a e g)}\) |
(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*Sqrt[f + g*x]*(a*d*e + (c*d^2 + a* e^2)*x + c*d*e*x^2)^(3/2)) - (4*g*((-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*Sqr t[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*g*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[f + g*x])))/(3*(c*d*f - a*e*g))
3.8.31.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ (n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] / ; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x] + Si mp[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))) Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e , f, g, n}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]
Time = 0.57 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (8 g^{2} x^{2} c^{2} d^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -c^{2} d^{2} f^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{3}}\) | \(121\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \sqrt {g x +f}\, \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(169\) |
int((e*x+d)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, method=_RETURNVERBOSE)
-2/3/(e*x+d)^(1/2)/(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(8*c^2*d^2*g^ 2*x^2+12*a*c*d*e*g^2*x+4*c^2*d^2*f*g*x+3*a^2*e^2*g^2+6*a*c*d*e*f*g-c^2*d^2 *f^2)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^3
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (170) = 340\).
Time = 0.53 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.44 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} - c^{2} d^{2} f^{2} + 6 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \, {\left (c^{2} d^{2} f g + 3 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a^{2} c^{3} d^{4} e^{2} f^{4} - 3 \, a^{3} c^{2} d^{3} e^{3} f^{3} g + 3 \, a^{4} c d^{2} e^{4} f^{2} g^{2} - a^{5} d e^{5} f g^{3} + {\left (c^{5} d^{5} e f^{3} g - 3 \, a c^{4} d^{4} e^{2} f^{2} g^{2} + 3 \, a^{2} c^{3} d^{3} e^{3} f g^{3} - a^{3} c^{2} d^{2} e^{4} g^{4}\right )} x^{4} + {\left (c^{5} d^{5} e f^{4} + {\left (c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} f^{3} g - 3 \, {\left (a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} + {\left (3 \, a^{2} c^{3} d^{4} e^{2} + 5 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} - {\left (a^{3} c^{2} d^{3} e^{3} + 2 \, a^{4} c d e^{5}\right )} g^{4}\right )} x^{3} + {\left ({\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} - {\left (a c^{4} d^{5} e + 5 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g - 3 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} + {\left (5 \, a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f g^{3} - {\left (2 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} g^{4}\right )} x^{2} - {\left (a^{5} d e^{5} g^{4} - {\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{4} + {\left (5 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{3} g - 3 \, {\left (a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f^{2} g^{2} - {\left (a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f g^{3}\right )} x\right )}} \]
integrate((e*x+d)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="fricas")
2/3*(8*c^2*d^2*g^2*x^2 - c^2*d^2*f^2 + 6*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 4*( c^2*d^2*f*g + 3*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x )*sqrt(e*x + d)*sqrt(g*x + f)/(a^2*c^3*d^4*e^2*f^4 - 3*a^3*c^2*d^3*e^3*f^3 *g + 3*a^4*c*d^2*e^4*f^2*g^2 - a^5*d*e^5*f*g^3 + (c^5*d^5*e*f^3*g - 3*a*c^ 4*d^4*e^2*f^2*g^2 + 3*a^2*c^3*d^3*e^3*f*g^3 - a^3*c^2*d^2*e^4*g^4)*x^4 + ( c^5*d^5*e*f^4 + (c^5*d^6 - a*c^4*d^4*e^2)*f^3*g - 3*(a*c^4*d^5*e + a^2*c^3 *d^3*e^3)*f^2*g^2 + (3*a^2*c^3*d^4*e^2 + 5*a^3*c^2*d^2*e^4)*f*g^3 - (a^3*c ^2*d^3*e^3 + 2*a^4*c*d*e^5)*g^4)*x^3 + ((c^5*d^6 + 2*a*c^4*d^4*e^2)*f^4 - (a*c^4*d^5*e + 5*a^2*c^3*d^3*e^3)*f^3*g - 3*(a^2*c^3*d^4*e^2 - a^3*c^2*d^2 *e^4)*f^2*g^2 + (5*a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f*g^3 - (2*a^4*c*d^2*e^4 + a^5*e^6)*g^4)*x^2 - (a^5*d*e^5*g^4 - (2*a*c^4*d^5*e + a^2*c^3*d^3*e^3)* f^4 + (5*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g - 3*(a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f^2*g^2 - (a^4*c*d^2*e^4 - a^5*e^6)*f*g^3)*x)
Timed out. \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
integrate((e*x+d)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="maxima")
integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*( g*x + f)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1923 vs. \(2 (170) = 340\).
Time = 0.50 (sec) , antiderivative size = 1923, normalized size of antiderivative = 9.91 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
integrate((e*x+d)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="giac")
2/3*(6*sqrt(c*d*g)*g^3/((c^2*d^2*e*f^2*abs(g) - 2*a*c*d*e^2*f*g*abs(g) + a ^2*e^3*g^2*abs(g))*(c*d*e^2*f*g - a*e^3*g^2 + (sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) - sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)* e*g - d*e*g)*c*d*g))^2)) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(5*(c^5*d^5 *e*f^2*g^4*abs(g) - 2*a*c^4*d^4*e^2*f*g^5*abs(g) + a^2*c^3*d^3*e^3*g^6*abs (g))*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^6*d^6*e^4*f^5*g^2 - 5*a*c^5*d^5*e^ 5*f^4*g^3 + 10*a^2*c^4*d^4*e^6*f^3*g^4 - 10*a^3*c^3*d^3*e^7*f^2*g^5 + 5*a^ 4*c^2*d^2*e^8*f*g^6 - a^5*c*d*e^9*g^7) - 6*(c^5*d^5*e^3*f^3*g^4*abs(g) - 3 *a*c^4*d^4*e^4*f^2*g^5*abs(g) + 3*a^2*c^3*d^3*e^5*f*g^6*abs(g) - a^3*c^2*d ^2*e^6*g^7*abs(g))/(c^6*d^6*e^4*f^5*g^2 - 5*a*c^5*d^5*e^5*f^4*g^3 + 10*a^2 *c^4*d^4*e^6*f^3*g^4 - 10*a^3*c^3*d^3*e^7*f^2*g^5 + 5*a^4*c^2*d^2*e^8*f*g^ 6 - a^5*c*d*e^9*g^7))/((c*d*e^2*f*g - a*e^3*g^2 - (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d* e*g)*c*d*g)))*e^3 - 2/3*(sqrt(e^2*f - d*e*g)*c^3*d^3*e^2*f^2*g^2 + 4*sqrt( e^2*f - d*e*g)*c^3*d^4*e*f*g^3 - 6*sqrt(e^2*f - d*e*g)*a*c^2*d^2*e^3*f*g^3 - 5*sqrt(e^2*f - d*e*g)*c^3*d^5*g^4 + 6*sqrt(e^2*f - d*e*g)*a*c^2*d^3*e^2 *g^4 - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*g - sqrt (-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^2*d^3*e*f*g^2 + 3*sqrt(-c*d^2*e*g ^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c*d*e^3*f*g^2 + 5*sqrt(-c*d^2*e*g^2 + a*e^3* g^2)*sqrt(c*d*g)*c^2*d^4*g^3 - 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*...
Time = 13.94 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g-2\,c^2\,d^2\,f^2\right )}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {8\,g\,x\,\left (3\,a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^3\,\sqrt {f+g\,x}+\frac {a^2\,e\,\sqrt {f+g\,x}}{c^2\,d}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}} \]
-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*g^2*x^2*(d + e*x)^(1/ 2))/(3*e*(a*e*g - c*d*f)^3) + ((d + e*x)^(1/2)*(6*a^2*e^2*g^2 - 2*c^2*d^2* f^2 + 12*a*c*d*e*f*g))/(3*c^2*d^2*e*(a*e*g - c*d*f)^3) + (8*g*x*(3*a*e*g + c*d*f)*(d + e*x)^(1/2))/(3*c*d*e*(a*e*g - c*d*f)^3)))/(x^3*(f + g*x)^(1/2 ) + (a^2*e*(f + g*x)^(1/2))/(c^2*d) + (x^2*(f + g*x)^(1/2)*(2*a*e^2 + c*d^ 2))/(c*d*e) + (a*x*(f + g*x)^(1/2)*(a*e^2 + 2*c*d^2))/(c^2*d^2))