Integrand size = 48, antiderivative size = 129 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{7/2}} \]
2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)/(e*x+d)^(7/2)/( g*x+f)^(9/2)+4/63*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d* f)^2/(e*x+d)^(7/2)/(g*x+f)^(7/2)
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{7/2} (-7 a e g+c d (9 f+2 g x))}{63 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}} \]
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*( f + g*x)^(11/2)),x]
(2*((a*e + c*d*x)*(d + e*x))^(7/2)*(-7*a*e*g + c*d*(9*f + 2*g*x)))/(63*(c* d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(9/2))
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1254, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}}dx}{9 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1248 |
| \(\displaystyle \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(7/2))/(63*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(7/2))
3.8.58.3.1 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)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] - Simp[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))) Int[(d + e*x)^m *(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[n, -1 ] && IntegerQ[2*p]
Time = 0.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +7 a e g -9 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (g x +f \right )^{\frac {9}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}\) | \(99\) |
| default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 c^{3} d^{3} g \,x^{3}+3 a \,c^{2} d^{2} e g \,x^{2}-9 c^{3} d^{3} f \,x^{2}+12 a^{2} c d \,e^{2} g x -18 a \,c^{2} d^{2} e f x +7 a^{3} e^{3} g -9 a^{2} c d \,e^{2} f \right ) \left (c d x +a e \right )}{63 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {9}{2}} \left (a e g -c d f \right )^{2}}\) | \(136\) |
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(11/2),x ,method=_RETURNVERBOSE)
-2/63*(c*d*x+a*e)*(-2*c*d*g*x+7*a*e*g-9*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+ a*d*e)^(5/2)/(g*x+f)^(9/2)/(a^2*e^2*g^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(e*x+d) ^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (113) = 226\).
Time = 0.60 (sec) , antiderivative size = 639, normalized size of antiderivative = 4.95 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\frac {2 \, {\left (2 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 7 \, a^{4} e^{4} g + {\left (9 \, c^{4} d^{4} f - a c^{3} d^{3} e g\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{3} e f - 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{2} e^{2} f - 19 \, a^{3} c d e^{3} g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{63 \, {\left (c^{2} d^{3} f^{7} - 2 \, a c d^{2} e f^{6} g + a^{2} d e^{2} f^{5} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{5} - 2 \, a c d e^{2} f g^{6} + a^{2} e^{3} g^{7}\right )} x^{6} + {\left (5 \, c^{2} d^{2} e f^{3} g^{4} + a^{2} d e^{2} g^{7} + {\left (c^{2} d^{3} - 10 \, a c d e^{2}\right )} f^{2} g^{5} - {\left (2 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} f g^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{2} d^{2} e f^{4} g^{3} + a^{2} d e^{2} f g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{3} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{5}\right )} x^{4} + 10 \, {\left (c^{2} d^{2} e f^{5} g^{2} + a^{2} d e^{2} f^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{4}\right )} x^{3} + 5 \, {\left (c^{2} d^{2} e f^{6} g + 2 \, a^{2} d e^{2} f^{3} g^{4} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{5} g^{2} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{7} + 5 \, a^{2} d e^{2} f^{4} g^{3} + {\left (5 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{6} g - {\left (10 \, a c d^{2} e - a^{2} e^{3}\right )} f^{5} g^{2}\right )} x\right )}} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 1/2),x, algorithm="fricas")
2/63*(2*c^4*d^4*g*x^4 + 9*a^3*c*d*e^3*f - 7*a^4*e^4*g + (9*c^4*d^4*f - a*c ^3*d^3*e*g)*x^3 + 3*(9*a*c^3*d^3*e*f - 5*a^2*c^2*d^2*e^2*g)*x^2 + (27*a^2* c^2*d^2*e^2*f - 19*a^3*c*d*e^3*g)*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)/(c^2*d^3*f^7 - 2*a*c*d^2*e*f^6*g + a^2* d*e^2*f^5*g^2 + (c^2*d^2*e*f^2*g^5 - 2*a*c*d*e^2*f*g^6 + a^2*e^3*g^7)*x^6 + (5*c^2*d^2*e*f^3*g^4 + a^2*d*e^2*g^7 + (c^2*d^3 - 10*a*c*d*e^2)*f^2*g^5 - (2*a*c*d^2*e - 5*a^2*e^3)*f*g^6)*x^5 + 5*(2*c^2*d^2*e*f^4*g^3 + a^2*d*e^ 2*f*g^6 + (c^2*d^3 - 4*a*c*d*e^2)*f^3*g^4 - 2*(a*c*d^2*e - a^2*e^3)*f^2*g^ 5)*x^4 + 10*(c^2*d^2*e*f^5*g^2 + a^2*d*e^2*f^2*g^5 + (c^2*d^3 - 2*a*c*d*e^ 2)*f^4*g^3 - (2*a*c*d^2*e - a^2*e^3)*f^3*g^4)*x^3 + 5*(c^2*d^2*e*f^6*g + 2 *a^2*d*e^2*f^3*g^4 + 2*(c^2*d^3 - a*c*d*e^2)*f^5*g^2 - (4*a*c*d^2*e - a^2* e^3)*f^4*g^3)*x^2 + (c^2*d^2*e*f^7 + 5*a^2*d*e^2*f^4*g^3 + (5*c^2*d^3 - 2* a*c*d*e^2)*f^6*g - (10*a*c*d^2*e - a^2*e^3)*f^5*g^2)*x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {11}{2}}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 1/2),x, algorithm="maxima")
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*( g*x + f)^(11/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (113) = 226\).
Time = 1.16 (sec) , antiderivative size = 1275, normalized size of antiderivative = 9.88 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=\text {Too large to display} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 1/2),x, algorithm="giac")
2/63*(9*sqrt(-c*d^2*e + a*e^3)*c^4*d^7*e*f*abs(c)*abs(d) - 27*sqrt(-c*d^2* e + a*e^3)*a*c^3*d^5*e^3*f*abs(c)*abs(d) + 27*sqrt(-c*d^2*e + a*e^3)*a^2*c ^2*d^3*e^5*f*abs(c)*abs(d) - 9*sqrt(-c*d^2*e + a*e^3)*a^3*c*d*e^7*f*abs(c) *abs(d) - 2*sqrt(-c*d^2*e + a*e^3)*c^4*d^8*g*abs(c)*abs(d) - sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2*g*abs(c)*abs(d) + 15*sqrt(-c*d^2*e + a*e^3)*a^2*c^ 2*d^4*e^4*g*abs(c)*abs(d) - 19*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6*g*abs( c)*abs(d) + 7*sqrt(-c*d^2*e + a*e^3)*a^4*e^8*g*abs(c)*abs(d))/(sqrt(c^2*d^ 2*e^2*f - c^2*d^3*e*g)*c^2*d^2*e^4*f^6 - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e* g)*c^2*d^3*e^3*f^5*g - 2*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d*e^5*f^5*g + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^4*e^2*f^4*g^2 + 8*sqrt(c^2*d^ 2*e^2*f - c^2*d^3*e*g)*a*c*d^2*e^4*f^4*g^2 + sqrt(c^2*d^2*e^2*f - c^2*d^3* e*g)*a^2*e^6*f^4*g^2 - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^5*e*f^3*g ^3 - 12*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^3*e^3*f^3*g^3 - 4*sqrt(c^2 *d^2*e^2*f - c^2*d^3*e*g)*a^2*d*e^5*f^3*g^3 + sqrt(c^2*d^2*e^2*f - c^2*d^3 *e*g)*c^2*d^6*f^2*g^4 + 8*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^4*e^2*f^ 2*g^4 + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*d^2*e^4*f^2*g^4 - 2*sqrt(c ^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^5*e*f*g^5 - 4*sqrt(c^2*d^2*e^2*f - c^2*d ^3*e*g)*a^2*d^3*e^3*f*g^5 + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*d^4*e^2* g^6) + 2/63*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*(2*(c^10*d^10*e^8*f^ 2*g^5*abs(c)*abs(d) - 2*a*c^9*d^9*e^9*f*g^6*abs(c)*abs(d) + a^2*c^8*d^8...
Time = 13.08 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^3\,e^3\,\left (7\,a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^4\,d^4\,x^4}{63\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (19\,a\,e\,g-27\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (5\,a\,e\,g-9\,c\,d\,f\right )}{21\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]
-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*a^3*e^3*(7*a*e*g - 9*c *d*f))/(63*g^4*(a*e*g - c*d*f)^2) - (4*c^4*d^4*x^4)/(63*g^3*(a*e*g - c*d*f )^2) + (2*c^3*d^3*x^3*(a*e*g - 9*c*d*f))/(63*g^4*(a*e*g - c*d*f)^2) + (2*a ^2*c*d*e^2*x*(19*a*e*g - 27*c*d*f))/(63*g^4*(a*e*g - c*d*f)^2) + (2*a*c^2* d^2*e*x^2*(5*a*e*g - 9*c*d*f))/(21*g^4*(a*e*g - c*d*f)^2)))/(x^4*(f + g*x) ^(1/2)*(d + e*x)^(1/2) + (f^4*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^4 + (4*f* x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (4*f^3*x*(f + g*x)^(1/2)*(d + e*x )^(1/2))/g^3 + (6*f^2*x^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2)