\(\int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [729]

Optimal result

Integrand size = 48, antiderivative size = 63 \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \]

[Out]

-2/3*(e*x+d)^(3/2)*(g*x+f)^(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)

Rubi [A] (verified)

Time = 0.04 (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} \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[In]

Int[((d + e*x)^(5/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

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 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) ((a e+c d x) (d+e x))^{3/2}} \]

[In]

Integrate[((d + e*x)^(5/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {2 \left (g x +f \right )^{\frac {3}{2}} \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )}\)	\(55\)
gosper	\(\frac {2 \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a e g -c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(63\)

[In]

int((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (55) = 110\).

Time = 0.32 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.06 \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} {\left (g x + f\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} c d^{2} e^{2} f - a^{3} d e^{3} g + {\left (c^{3} d^{3} e f - a c^{2} d^{2} e^{2} g\right )} x^{3} + {\left ({\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} g\right )} x^{2} + {\left ({\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f - {\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g\right )} x\right )}} \]

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((e*x+d)**(5/2)*(g*x+f)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (55) = 110\).

Time = 0.45 (sec) , antiderivative size = 316, normalized size of antiderivative = 5.02 \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )}^{\frac {3}{2}} c d e^{2} g^{4}}{3 \, {\left (c^{2} d^{2} e^{2} f {\left | g \right |} - a c d e^{3} g {\left | g \right |}\right )} {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )} \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} - \frac {2 \, {\left (\sqrt {e^{2} f - d e g} e f g^{2} - \sqrt {e^{2} f - d e g} d g^{3}\right )}}{3 \, {\left (\sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{2} d^{3} f {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c d e^{2} f {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c d^{2} e g {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} e^{3} g {\left | g \right |}\right )}} \]

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.87 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.68 \[ \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\left (\frac {2\,f\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}+\frac {2\,g\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}} \]

[In]

int(((f + g*x)^(1/2)*(d + e*x)^(5/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

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