\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx\) [757]

Optimal result

Integrand size = 48, antiderivative size = 63 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}} \]

[Out]

2/7*(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)^(7/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {874} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \]

[In]

Int[(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)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

Rule 874

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] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}} \]

[In]

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)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {5}{2}}}\)	\(63\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c^{2} d^{2} x^{2}+2 a c d e x +e^{2} a^{2}\right ) \left (c d x +a e \right )}{7 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )}\)	\(78\)

[In]

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)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/7/(g*x+f)^(7/2)*(c*d*x+a*e)/(a*e*g-c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (55) = 110\).

Time = 0.44 (sec) , antiderivative size = 299, normalized size of antiderivative = 4.75 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{7 \, {\left (c d^{2} f^{5} - a d e f^{4} g + {\left (c d e f g^{4} - a e^{2} g^{5}\right )} x^{5} + {\left (4 \, c d e f^{2} g^{3} - a d e g^{5} + {\left (c d^{2} - 4 \, a e^{2}\right )} f g^{4}\right )} x^{4} + 2 \, {\left (3 \, c d e f^{3} g^{2} - 2 \, a d e f g^{4} + {\left (2 \, c d^{2} - 3 \, a e^{2}\right )} f^{2} g^{3}\right )} x^{3} + 2 \, {\left (2 \, c d e f^{4} g - 3 \, a d e f^{2} g^{3} + {\left (3 \, c d^{2} - 2 \, a e^{2}\right )} f^{3} g^{2}\right )} x^{2} + {\left (c d e f^{5} - 4 \, a d e f^{3} g^{2} + {\left (4 \, c d^{2} - a e^{2}\right )} f^{4} g\right )} x\right )}} \]

[In]

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)^(9/2),x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*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*d^2*f^5 - a*d*e*f^4*g + (c*d*e*f*g^4 - a*e^2*g^5)*x^5 + (4*c*d*e*f^2*g^3 - a*d*
e*g^5 + (c*d^2 - 4*a*e^2)*f*g^4)*x^4 + 2*(3*c*d*e*f^3*g^2 - 2*a*d*e*f*g^4 + (2*c*d^2 - 3*a*e^2)*f^2*g^3)*x^3 +
2*(2*c*d*e*f^4*g - 3*a*d*e*f^2*g^3 + (3*c*d^2 - 2*a*e^2)*f^3*g^2)*x^2 + (c*d*e*f^5 - 4*a*d*e*f^3*g^2 + (4*c*d
^2 - a*e^2)*f^4*g)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\text {Timed out} \]

[In]

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)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]

[In]

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)^(9/2),x, algorithm="maxima")

[Out]

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)^(9/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (55) = 110\).

Time = 0.78 (sec) , antiderivative size = 602, normalized size of antiderivative = 9.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=\frac {2 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} {\left | c \right |} {\left | d \right |} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} {\left | c \right |} {\left | d \right |} + 3 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} {\left | c \right |} {\left | d \right |} - \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6} {\left | c \right |} {\left | d \right |}\right )}}{7 \, {\left (\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d e^{3} f^{4} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{2} e^{2} f^{3} g - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a e^{4} f^{3} g + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{3} e f^{2} g^{2} + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d e^{3} f^{2} g^{2} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c d^{4} f g^{3} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d^{2} e^{2} f g^{3} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a d^{3} e g^{4}\right )}} + \frac {2 \, {\left (c^{8} d^{8} e^{6} f^{2} g^{3} {\left | c \right |} {\left | d \right |} - 2 \, a c^{7} d^{7} e^{7} f g^{4} {\left | c \right |} {\left | d \right |} + a^{2} c^{6} d^{6} e^{8} g^{5} {\left | c \right |} {\left | d \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}}{7 \, {\left (c^{3} d^{3} e^{6} f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{7} f^{2} g^{4} + 3 \, a^{2} c d e^{8} f g^{5} - a^{3} e^{9} g^{6}\right )} {\left (c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g\right )}^{\frac {7}{2}}} \]

[In]

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)^(9/2),x, algorithm="giac")

[Out]

2/7*(sqrt(-c*d^2*e + a*e^3)*c^3*d^6*abs(c)*abs(d) - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2*abs(c)*abs(d) + 3*s
qrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4*abs(c)*abs(d) - sqrt(-c*d^2*e + a*e^3)*a^3*e^6*abs(c)*abs(d))/(sqrt(c^2*d^
2*e^2*f - c^2*d^3*e*g)*c*d*e^3*f^4 - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c*d^2*e^2*f^3*g - sqrt(c^2*d^2*e^2*f
- c^2*d^3*e*g)*a*e^4*f^3*g + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c*d^3*e*f^2*g^2 + 3*sqrt(c^2*d^2*e^2*f - c^2*
d^3*e*g)*a*d*e^3*f^2*g^2 - sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c*d^4*f*g^3 - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)
*a*d^2*e^2*f*g^3 + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*d^3*e*g^4) + 2/7*(c^8*d^8*e^6*f^2*g^3*abs(c)*abs(d) - 2
*a*c^7*d^7*e^7*f*g^4*abs(c)*abs(d) + a^2*c^6*d^6*e^8*g^5*abs(c)*abs(d))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7
/2)/((c^3*d^3*e^6*f^3*g^3 - 3*a*c^2*d^2*e^7*f^2*g^4 + 3*a^2*c*d*e^8*f*g^5 - a^3*e^9*g^6)*(c^2*d^2*e^2*f - a*c*
d*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)^(7/2))

Mupad [B] (verification not implemented)

Time = 12.82 (sec) , antiderivative size = 325, normalized size of antiderivative = 5.16 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^3\,e^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {2\,c^3\,d^3\,x^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a^2\,c\,d\,e^2\,x}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a\,c^2\,d^2\,e\,x^2}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (7\,c\,d\,f^4-7\,a\,e\,f^3\,g\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {x^2\,\sqrt {f+g\,x}\,\left (21\,a\,e\,f\,g^3-21\,c\,d\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}-\frac {x\,\sqrt {f+g\,x}\,\left (21\,c\,d\,f^3\,g-21\,a\,e\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(9/2)*(d + e*x)^(5/2)),x)

[Out]

-((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^4 - 7*c*d*f*g^3) + (2*c^3*d^3*x^3)/(7*a*
e*g^4 - 7*c*d*f*g^3) + (6*a^2*c*d*e^2*x)/(7*a*e*g^4 - 7*c*d*f*g^3) + (6*a*c^2*d^2*e*x^2)/(7*a*e*g^4 - 7*c*d*f*
g^3)))/(x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2) - ((f + g*x)^(1/2)*(7*c*d*f^4 - 7*a*e*f^3*g)*(d + e*x)^(1/2))/(7*a
*e*g^4 - 7*c*d*f*g^3) + (x^2*(f + g*x)^(1/2)*(21*a*e*f*g^3 - 21*c*d*f^2*g^2)*(d + e*x)^(1/2))/(7*a*e*g^4 - 7*c
*d*f*g^3) - (x*(f + g*x)^(1/2)*(21*c*d*f^3*g - 21*a*e*f^2*g^2)*(d + e*x)^(1/2))/(7*a*e*g^4 - 7*c*d*f*g^3))