3.7.78 \(\int \frac {(d+e x)^{5/2}}{(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [678]

Optimal. Leaf size=342 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {35 c^2 d^2 g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{9/2}} \]

[Out]

-2/3*(e*x+d)^(3/2)/(-a*e*g+c*d*f)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+35/4*c^2*d^2*g^(3/2)*arcta
n(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))/(-a*e*g+c*d*f)^(9/2)+14/
3*g*(e*x+d)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/6*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)^2/(e*x+d)^(1/2)+35/4*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/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {882, 886, 888, 211} \begin {gather*} \frac {35 c^2 d^2 g^{3/2} \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 (c d f-a e g)^{9/2}}+\frac {35 c d g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt {d+e x} (f+g x) (c d f-a e g)^4}+\frac {35 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^3}+\frac {14 g \sqrt {d+e x}}{3 (f+g x)^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)^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)^3*(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)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (14*g*Sqr
t[d + e*x])/(3*(c*d*f - a*e*g)^2*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*g^2*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^2) + (35*c*d*g^2*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f + g*x)) + (35*c^2*d^2*g^(3/2)*ArcTan[(Sq
rt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(4*(c*d*f - a*e*g)^(9
/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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]

Rule 888

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, 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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{(f+g x)^3 \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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(7 g) \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \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}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 g^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \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)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {\left (35 c d g^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 (c d f-a e g)^3}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {\left (35 c^2 d^2 g^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 (c d f-a e g)^4}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {\left (35 c^2 d^2 e^2 g^2\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{4 (c d f-a e g)^4}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {35 c^2 d^2 g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 240, normalized size = 0.70 \begin {gather*} \frac {c^2 d^2 \sqrt {d+e x} \left (\frac {-6 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (13 f+7 g x)+2 a c^2 d^2 e g \left (40 f^2+119 f g x+70 g^2 x^2\right )+c^3 d^3 \left (-8 f^3+56 f^2 g x+175 f g^2 x^2+105 g^3 x^3\right )}{c^2 d^2 (c d f-a e g)^4 (a e+c d x) (f+g x)^2}+\frac {105 g^{3/2} \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{9/2}}\right )}{12 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(c^2*d^2*Sqrt[d + e*x]*((-6*a^3*e^3*g^3 + 3*a^2*c*d*e^2*g^2*(13*f + 7*g*x) + 2*a*c^2*d^2*e*g*(40*f^2 + 119*f*g
*x + 70*g^2*x^2) + c^3*d^3*(-8*f^3 + 56*f^2*g*x + 175*f*g^2*x^2 + 105*g^3*x^3))/(c^2*d^2*(c*d*f - a*e*g)^4*(a*
e + c*d*x)*(f + g*x)^2) + (105*g^(3/2)*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g
]])/(c*d*f - a*e*g)^(9/2)))/(12*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(659\) vs. \(2(304)=608\).
time = 0.14, size = 660, normalized size = 1.93

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{4} x^{3} \sqrt {c d x +a e}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,g^{4} x^{2} \sqrt {c d x +a e}+210 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{3} x^{2} \sqrt {c d x +a e}+210 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e f \,g^{3} x \sqrt {c d x +a e}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g^{2} x \sqrt {c d x +a e}-105 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} g^{3} x^{3}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g^{2} \sqrt {c d x +a e}-140 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,g^{3} x^{2}-175 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f \,g^{2} x^{2}-21 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} g^{3} x -238 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e f \,g^{2} x -56 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{2} g x +6 \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}-39 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}-80 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +8 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{12 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{4} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(660\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/12*((c*d*x+a*e)*(e*x+d))^(1/2)*(105*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*g^4*x^3*(c
*d*x+a*e)^(1/2)+105*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*g^4*x^2*(c*d*x+a*e)^(1/2)
+210*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f*g^3*x^2*(c*d*x+a*e)^(1/2)+210*arctanh(g*(c
*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*f*g^3*x*(c*d*x+a*e)^(1/2)+105*arctanh(g*(c*d*x+a*e)^(1/2)
/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f^2*g^2*x*(c*d*x+a*e)^(1/2)-105*((a*e*g-c*d*f)*g)^(1/2)*c^3*d^3*g^3*x^3+105*
arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*f^2*g^2*(c*d*x+a*e)^(1/2)-140*((a*e*g-c*d*f)*
g)^(1/2)*a*c^2*d^2*e*g^3*x^2-175*((a*e*g-c*d*f)*g)^(1/2)*c^3*d^3*f*g^2*x^2-21*((a*e*g-c*d*f)*g)^(1/2)*a^2*c*d*
e^2*g^3*x-238*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2*e*f*g^2*x-56*((a*e*g-c*d*f)*g)^(1/2)*c^3*d^3*f^2*g*x+6*((a*e*g
-c*d*f)*g)^(1/2)*a^3*e^3*g^3-39*((a*e*g-c*d*f)*g)^(1/2)*a^2*c*d*e^2*f*g^2-80*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2
*e*f^2*g+8*((a*e*g-c*d*f)*g)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^4/(g*x+f)^2/((a*e*g-
c*d*f)*g)^(1/2)

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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)^3/(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)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (320) = 640\).
time = 6.27, size = 3043, normalized size = 8.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(105*(c^4*d^5*g^3*x^4 + 2*c^4*d^5*f*g^2*x^3 + c^4*d^5*f^2*g*x^2 + (a^2*c^2*d^2*g^3*x^3 + 2*a^2*c^2*d^2*f
*g^2*x^2 + a^2*c^2*d^2*f^2*g*x)*e^3 + (2*a*c^3*d^3*g^3*x^4 + 4*a*c^3*d^3*f*g^2*x^3 + 2*a^2*c^2*d^3*f*g^2*x + a
^2*c^2*d^3*f^2*g + (2*a*c^3*d^3*f^2*g + a^2*c^2*d^3*g^3)*x^2)*e^2 + (c^4*d^4*g^3*x^5 + 2*c^4*d^4*f*g^2*x^4 + 4
*a*c^3*d^4*f*g^2*x^2 + 2*a*c^3*d^4*f^2*g*x + (c^4*d^4*f^2*g + 2*a*c^3*d^4*g^3)*x^3)*e)*sqrt(-g/(c*d*f - a*g*e)
)*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d*f - a*g*e)*sqrt
(x*e + d)*sqrt(-g/(c*d*f - a*g*e)) + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e)/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*(
105*c^3*d^3*g^3*x^3 + 175*c^3*d^3*f*g^2*x^2 + 56*c^3*d^3*f^2*g*x - 8*c^3*d^3*f^3 - 6*a^3*g^3*e^3 + 3*(7*a^2*c*
d*g^3*x + 13*a^2*c*d*f*g^2)*e^2 + 2*(70*a*c^2*d^2*g^3*x^2 + 119*a*c^2*d^2*f*g^2*x + 40*a*c^2*d^2*f^2*g)*e)*sqr
t(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*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), 1/12*(105*(c^4*d^5*g^3*x^4 + 2*c^4*d^5*f*g^2*x^3 + c^4*d^5*f^2*g*x^2 + (a^2*c^2*d^2
*g^3*x^3 + 2*a^2*c^2*d^2*f*g^2*x^2 + a^2*c^2*d^2*f^2*g*x)*e^3 + (2*a*c^3*d^3*g^3*x^4 + 4*a*c^3*d^3*f*g^2*x^3 +
2*a^2*c^2*d^3*f*g^2*x + a^2*c^2*d^3*f^2*g + (2*a*c^3*d^3*f^2*g + a^2*c^2*d^3*g^3)*x^2)*e^2 + (c^4*d^4*g^3*x^5
+ 2*c^4*d^4*f*g^2*x^4 + 4*a*c^3*d^4*f*g^2*x^2 + 2*a*c^3*d^4*f^2*g*x + (c^4*d^4*f^2*g + 2*a*c^3*d^4*g^3)*x^3)*
e)*sqrt(g/(c*d*f - a*g*e))*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d*f - a*g*e)*sqrt(x*e + d)*s
qrt(g/(c*d*f - a*g*e))/(c*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) + (105*c^3*d^3*g^3*x^3 + 175*c^3*d^3*f
*g^2*x^2 + 56*c^3*d^3*f^2*g*x - 8*c^3*d^3*f^3 - 6*a^3*g^3*e^3 + 3*(7*a^2*c*d*g^3*x + 13*a^2*c*d*f*g^2)*e^2 + 2
*(70*a*c^2*d^2*g^3*x^2 + 119*a*c^2*d^2*f*g^2*x + 40*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(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)**3/(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)^3/(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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{5/2}}{{\left (f+g\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/((f + g*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)

[Out]

int((d + e*x)^(5/2)/((f + g*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), x)

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