\(\int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [730]

Optimal result

Integrand size = 48, antiderivative size = 128 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x} \sqrt {f+g x}}{3 (c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, 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 = {882, 874} \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {4 g \sqrt {d+e x} \sqrt {f+g x}}{3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[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)*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*g*Sq
rt[d + e*x]*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^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]

Rule 882

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]
+ Dist[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] && 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] && LtQ[p, -1] && RationalQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{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}}-\frac {(2 g) \int \frac {(d+e x)^{3/2}}{\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}} \, dx}{3 (c d f-a e g)} \\ & = -\frac {2 (d+e x)^{3/2} \sqrt {f+g x}}{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}}+\frac {4 g \sqrt {d+e x} \sqrt {f+g x}}{3 (c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} \sqrt {f+g x} (-3 a e g+c d (f-2 g x))}{3 (c d f-a e g)^2 ((a e+c d x) (d+e x))^{3/2}} \]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (2 c d g x +3 a e g -c d f \right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{2}}\)	\(72\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (c d x +a e \right ) \left (2 c d g x +3 a e g -c d f \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(99\)

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (112) = 224\).

Time = 0.44 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x - c d f + 3 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a^{2} c^{2} d^{3} e^{2} f^{2} - 2 \, a^{3} c d^{2} e^{3} f g + a^{4} d e^{4} g^{2} + {\left (c^{4} d^{4} e f^{2} - 2 \, a c^{3} d^{3} e^{2} f g + a^{2} c^{2} d^{2} e^{3} g^{2}\right )} x^{3} + {\left ({\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} - 2 \, {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3}\right )} f g + {\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 2 \, {\left (2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g + {\left (2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{2}\right )} x\right )}} \]

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

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (112) = 224\).

Time = 0.35 (sec) , antiderivative size = 552, normalized size of antiderivative = 4.31 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c^{2} d^{2} g^{4}}{c^{3} d^{3} e^{2} f^{2} {\left | g \right |} - 2 \, a c^{2} d^{2} e^{3} f g {\left | g \right |} + a^{2} c d e^{4} g^{2} {\left | g \right |}} - \frac {3 \, {\left (c^{2} d^{2} e^{2} f g^{4} - a c d e^{3} g^{5}\right )}}{c^{3} d^{3} e^{2} f^{2} {\left | g \right |} - 2 \, a c^{2} d^{2} e^{3} f g {\left | g \right |} + a^{2} c d e^{4} g^{2} {\left | g \right |}}\right )} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} e^{2}}{3 \, {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )} \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} - \frac {2 \, {\left (\sqrt {e^{2} f - d e g} c d e f g^{2} + 2 \, \sqrt {e^{2} f - d e g} c d^{2} g^{3} - 3 \, \sqrt {e^{2} f - d e g} a e^{2} g^{3}\right )}}{3 \, {\left (\sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{3} d^{4} f^{2} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{2} e^{2} f^{2} {\left | g \right |} - 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c^{2} d^{3} e f g {\left | g \right |} + 2 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d e^{3} f g {\left | g \right |} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} c d^{2} e^{2} g^{2} {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{3} e^{4} g^{2} {\left | g \right |}\right )}} \]

[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*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*c^2*d^2*g^4/(c^3*d^3*e^2*f^2*abs(g) - 2*a*c^2*d^2*e^3*f*g*abs(g) + a^2
*c*d*e^4*g^2*abs(g)) - 3*(c^2*d^2*e^2*f*g^4 - a*c*d*e^3*g^5)/(c^3*d^3*e^2*f^2*abs(g) - 2*a*c^2*d^2*e^3*f*g*abs
(g) + a^2*c*d*e^4*g^2*abs(g)))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*e^2/((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*(sqr
t(e^2*f - d*e*g)*c*d*e*f*g^2 + 2*sqrt(e^2*f - d*e*g)*c*d^2*g^3 - 3*sqrt(e^2*f - d*e*g)*a*e^2*g^3)/(sqrt(-c*d^2
*e*g^2 + a*e^3*g^2)*c^3*d^4*f^2*abs(g) - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^2*d^2*e^2*f^2*abs(g) - 2*sqrt(-c*d
^2*e*g^2 + a*e^3*g^2)*a*c^2*d^3*e*f*g*abs(g) + 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c*d*e^3*f*g*abs(g) + sqrt(
-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c*d^2*e^2*g^2*abs(g) - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*e^4*g^2*abs(g))

Mupad [B] (verification not implemented)

Time = 13.62 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.92 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {\left (2\,c\,d\,f^2-6\,a\,e\,f\,g\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {x\,\left (6\,a\,e\,g^2+2\,c\,d\,f\,g\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^3\,\sqrt {f+g\,x}+\frac {a^2\,e\,\sqrt {f+g\,x}}{c^2\,d}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}} \]

[In]

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

[Out]

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