3.8.32 \(\int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [732]

Optimal. Leaf size=260 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2/3*(e*x+d)^(3/2)/(-a*e*g+c*d*f)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+4*g*(e*x+d)^(1/2)/(-a*
e*g+c*d*f)^2/(g*x+f)^(3/2)/(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/(g*x+f)^(3/2)/(e*x+d)^(1/2)+32/3*c*d*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)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {882, 886, 874} \begin {gather*} \frac {32 c d g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}+\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {4 g \sqrt {d+e x}}{(f+g x)^{3/2} \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}}{3 (f+g x)^{3/2} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(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))/(3*(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*g*
Sqrt[d + e*x])/((c*d*f - a*e*g)^2*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (16*g^2*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c*d*g^2*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

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]

Rule 886

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] - Dist[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] && 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[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \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) (f+g x)^{3/2} \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}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d f-a e g}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \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}}{(c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 g^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{(c d f-a e g)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \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}}{(c d f-a e g)^2 (f+g x)^{3/2} \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} (f+g x)^{3/2}}+\frac {\left (16 c d g^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 (c d f-a e g)^3}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \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}}{(c d f-a e g)^2 (f+g x)^{3/2} \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} (f+g x)^{3/2}}+\frac {32 c d 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)^4 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 138, normalized size = 0.53 \begin {gather*} -\frac {2 (a e+c d x)^4 (d+e x)^{5/2} \left (g^3-\frac {9 c d g^2 (f+g x)}{a e+c d x}-\frac {9 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{3 (c d f-a e g)^4 ((a e+c d x) (d+e x))^{5/2} (f+g x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*e + c*d*x)^4*(d + e*x)^(5/2)*(g^3 - (9*c*d*g^2*(f + g*x))/(a*e + c*d*x) - (9*c^2*d^2*g*(f + g*x)^2)/(a*
e + c*d*x)^2 + (c^3*d^3*(f + g*x)^3)/(a*e + c*d*x)^3))/(3*(c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*x))^(5/2)*(f
+ g*x)^(3/2))

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Maple [A]
time = 0.13, size = 191, normalized size = 0.73

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{3} c^{3} d^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{4}}\)	\(191\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(258\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(g*x+f)^(5/2)/(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)*(-16*c^3*d^3*g^3*x^3-24*a*c^2*d^2*e*g^3*x^2-24*c^
3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x-36*a*c^2*d^2*e*f*g^2*x-6*c^3*d^3*f^2*g*x+a^3*e^3*g^3-9*a^2*c*d*e^2*f*g^2-9
*a*c^2*d^2*e*f^2*g+c^3*d^3*f^3)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^4

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (242) = 484\).
time = 6.49, size = 1108, normalized size = 4.26 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 24 \, c^{3} d^{3} f g^{2} x^{2} + 6 \, c^{3} d^{3} f^{2} g x - c^{3} d^{3} f^{3} - a^{3} g^{3} e^{3} + 3 \, {\left (2 \, a^{2} c d g^{3} x + 3 \, a^{2} c d f g^{2}\right )} e^{2} + 3 \, {\left (8 \, a c^{2} d^{2} g^{3} x^{2} + 12 \, a c^{2} d^{2} f g^{2} x + 3 \, a c^{2} d^{2} f^{2} g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{3 \, {\left (c^{6} d^{7} f^{4} g^{2} x^{4} + 2 \, c^{6} d^{7} f^{5} g x^{3} + c^{6} d^{7} f^{6} x^{2} + {\left (a^{6} g^{6} x^{3} + 2 \, a^{6} f g^{5} x^{2} + a^{6} f^{2} g^{4} x\right )} e^{7} + {\left (2 \, a^{5} c d g^{6} x^{4} + a^{6} d f^{2} g^{4} - {\left (6 \, a^{5} c d f^{2} g^{4} - a^{6} d g^{6}\right )} x^{2} - 2 \, {\left (2 \, a^{5} c d f^{3} g^{3} - a^{6} d f g^{5}\right )} x\right )} e^{6} + {\left (a^{4} c^{2} d^{2} g^{6} x^{5} - 6 \, a^{4} c^{2} d^{2} f g^{5} x^{4} + 4 \, a^{4} c^{2} d^{2} f^{3} g^{3} x^{2} - 4 \, a^{5} c d^{2} f^{3} g^{3} - {\left (9 \, a^{4} c^{2} d^{2} f^{2} g^{4} - 2 \, a^{5} c d^{2} g^{6}\right )} x^{3} + 6 \, {\left (a^{4} c^{2} d^{2} f^{4} g^{2} - a^{5} c d^{2} f^{2} g^{4}\right )} x\right )} e^{5} - {\left (4 \, a^{3} c^{3} d^{3} f g^{5} x^{5} - 6 \, a^{4} c^{2} d^{3} f^{4} g^{2} - {\left (4 \, a^{3} c^{3} d^{3} f^{2} g^{4} + a^{4} c^{2} d^{3} g^{6}\right )} x^{4} - 2 \, {\left (8 \, a^{3} c^{3} d^{3} f^{3} g^{3} - 3 \, a^{4} c^{2} d^{3} f g^{5}\right )} x^{3} - {\left (4 \, a^{3} c^{3} d^{3} f^{4} g^{2} - 9 \, a^{4} c^{2} d^{3} f^{2} g^{4}\right )} x^{2} + 4 \, {\left (a^{3} c^{3} d^{3} f^{5} g - a^{4} c^{2} d^{3} f^{3} g^{3}\right )} x\right )} e^{4} + {\left (6 \, a^{2} c^{4} d^{4} f^{2} g^{4} x^{5} - 4 \, a^{3} c^{3} d^{4} f^{5} g + 4 \, {\left (a^{2} c^{4} d^{4} f^{3} g^{3} - a^{3} c^{3} d^{4} f g^{5}\right )} x^{4} - {\left (9 \, a^{2} c^{4} d^{4} f^{4} g^{2} - 4 \, a^{3} c^{3} d^{4} f^{2} g^{4}\right )} x^{3} - 2 \, {\left (3 \, a^{2} c^{4} d^{4} f^{5} g - 8 \, a^{3} c^{3} d^{4} f^{3} g^{3}\right )} x^{2} + {\left (a^{2} c^{4} d^{4} f^{6} + 4 \, a^{3} c^{3} d^{4} f^{4} g^{2}\right )} x\right )} e^{3} - {\left (4 \, a c^{5} d^{5} f^{3} g^{3} x^{5} - 4 \, a^{2} c^{4} d^{5} f^{3} g^{3} x^{3} + 6 \, a^{2} c^{4} d^{5} f^{5} g x - a^{2} c^{4} d^{5} f^{6} + 6 \, {\left (a c^{5} d^{5} f^{4} g^{2} - a^{2} c^{4} d^{5} f^{2} g^{4}\right )} x^{4} - {\left (2 \, a c^{5} d^{5} f^{6} - 9 \, a^{2} c^{4} d^{5} f^{4} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{6} d^{6} f^{4} g^{2} x^{5} + 2 \, a c^{5} d^{6} f^{6} x + 2 \, {\left (c^{6} d^{6} f^{5} g - 2 \, a c^{5} d^{6} f^{3} g^{3}\right )} x^{4} + {\left (c^{6} d^{6} f^{6} - 6 \, a c^{5} d^{6} f^{4} g^{2}\right )} x^{3}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(16*c^3*d^3*g^3*x^3 + 24*c^3*d^3*f*g^2*x^2 + 6*c^3*d^3*f^2*g*x - c^3*d^3*f^3 - a^3*g^3*e^3 + 3*(2*a^2*c*d*
g^3*x + 3*a^2*c*d*f*g^2)*e^2 + 3*(8*a*c^2*d^2*g^3*x^2 + 12*a*c^2*d^2*f*g^2*x + 3*a*c^2*d^2*f^2*g)*e)*sqrt(c*d^
2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^6*d^7*f^4*g^2*x^4 + 2*c^6*d^7*f^5*g*x^3 + c^
6*d^7*f^6*x^2 + (a^6*g^6*x^3 + 2*a^6*f*g^5*x^2 + a^6*f^2*g^4*x)*e^7 + (2*a^5*c*d*g^6*x^4 + a^6*d*f^2*g^4 - (6*
a^5*c*d*f^2*g^4 - a^6*d*g^6)*x^2 - 2*(2*a^5*c*d*f^3*g^3 - a^6*d*f*g^5)*x)*e^6 + (a^4*c^2*d^2*g^6*x^5 - 6*a^4*c
^2*d^2*f*g^5*x^4 + 4*a^4*c^2*d^2*f^3*g^3*x^2 - 4*a^5*c*d^2*f^3*g^3 - (9*a^4*c^2*d^2*f^2*g^4 - 2*a^5*c*d^2*g^6)
*x^3 + 6*(a^4*c^2*d^2*f^4*g^2 - a^5*c*d^2*f^2*g^4)*x)*e^5 - (4*a^3*c^3*d^3*f*g^5*x^5 - 6*a^4*c^2*d^3*f^4*g^2 -
(4*a^3*c^3*d^3*f^2*g^4 + a^4*c^2*d^3*g^6)*x^4 - 2*(8*a^3*c^3*d^3*f^3*g^3 - 3*a^4*c^2*d^3*f*g^5)*x^3 - (4*a^3*
c^3*d^3*f^4*g^2 - 9*a^4*c^2*d^3*f^2*g^4)*x^2 + 4*(a^3*c^3*d^3*f^5*g - a^4*c^2*d^3*f^3*g^3)*x)*e^4 + (6*a^2*c^4
*d^4*f^2*g^4*x^5 - 4*a^3*c^3*d^4*f^5*g + 4*(a^2*c^4*d^4*f^3*g^3 - a^3*c^3*d^4*f*g^5)*x^4 - (9*a^2*c^4*d^4*f^4*
g^2 - 4*a^3*c^3*d^4*f^2*g^4)*x^3 - 2*(3*a^2*c^4*d^4*f^5*g - 8*a^3*c^3*d^4*f^3*g^3)*x^2 + (a^2*c^4*d^4*f^6 + 4*
a^3*c^3*d^4*f^4*g^2)*x)*e^3 - (4*a*c^5*d^5*f^3*g^3*x^5 - 4*a^2*c^4*d^5*f^3*g^3*x^3 + 6*a^2*c^4*d^5*f^5*g*x - a
^2*c^4*d^5*f^6 + 6*(a*c^5*d^5*f^4*g^2 - a^2*c^4*d^5*f^2*g^4)*x^4 - (2*a*c^5*d^5*f^6 - 9*a^2*c^4*d^5*f^4*g^2)*x
^2)*e^2 + (c^6*d^6*f^4*g^2*x^5 + 2*a*c^5*d^6*f^6*x + 2*(c^6*d^6*f^5*g - 2*a*c^5*d^6*f^3*g^3)*x^4 + (c^6*d^6*f^
6 - 6*a*c^5*d^6*f^4*g^2)*x^3)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 5.86, size = 416, normalized size = 1.60 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g\,x^2\,\left (a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {\sqrt {d+e\,x}\,\left (2\,a^3\,e^3\,g^3-18\,a^2\,c\,d\,e^2\,f\,g^2-18\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{3\,c^2\,d^2\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c\,d\,g^2\,x^3\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,x\,\sqrt {d+e\,x}\,\left (a^2\,e^2\,g^2+6\,a\,c\,d\,e\,f\,g+c^2\,d^2\,f^2\right )}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {x^2\,\sqrt {f+g\,x}\,\left (g\,a^2\,e^3+2\,g\,a\,c\,d^2\,e+2\,f\,a\,c\,d\,e^2+f\,c^2\,d^3\right )}{c^2\,d^2\,e\,g}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c^2\,d^2\,g}+\frac {a^2\,e\,f\,\sqrt {f+g\,x}}{c^2\,d\,g}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+2\,a\,g\,e^2\right )}{c\,d\,e\,g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/((f + g*x)^(5/2)*(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)*((16*g*x^2*(a*e*g + c*d*f)*(d + e*x)^(1/2))/(e*(a*e*g - c*d*f)^
4) - ((d + e*x)^(1/2)*(2*a^3*e^3*g^3 + 2*c^3*d^3*f^3 - 18*a*c^2*d^2*e*f^2*g - 18*a^2*c*d*e^2*f*g^2))/(3*c^2*d^
2*e*g*(a*e*g - c*d*f)^4) + (32*c*d*g^2*x^3*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^4) + (4*x*(d + e*x)^(1/2)*(a^
2*e^2*g^2 + c^2*d^2*f^2 + 6*a*c*d*e*f*g))/(c*d*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (x^2*(f + g*x)^(1
/2)*(a^2*e^3*g + c^2*d^3*f + 2*a*c*d*e^2*f + 2*a*c*d^2*e*g))/(c^2*d^2*e*g) + (a*x*(f + g*x)^(1/2)*(a*e^2*f + 2
*c*d^2*f + a*d*e*g))/(c^2*d^2*g) + (a^2*e*f*(f + g*x)^(1/2))/(c^2*d*g) + (x^3*(f + g*x)^(1/2)*(2*a*e^2*g + c*d
^2*g + c*d*e*f))/(c*d*e*g))

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