3.8.11 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx\) [711]

Optimal. Leaf size=463 \[ -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}} \]

[Out]

-1/12*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)/(g*x+f)^5-1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)/g/(e*x+d)^(5/2)/(g*x+f)^6+5/512*c^6*d^6*arctan(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))/g^(7/2)/(-a*e*g+c*d*f)^(7/2)-1/32*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/g^3/(g*x+f)^4/(e*x+d)^(1/2)+1/192*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)/(g*x
+f)^3/(e*x+d)^(1/2)+5/768*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+
d)^(1/2)+5/512*c^5*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.48, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \begin {gather*} \frac {5 c^6 d^6 \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 )}{512 g^{7/2} (c d f-a e g)^{7/2}}+\frac {5 c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 g^3 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{768 g^3 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 g^3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6} \end {gather*}

Antiderivative was successfully verified.

[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)^7),x]

[Out]

-1/32*(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]*(f + g*x)^4) + (c^3*d^3*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*g^3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c^4*d^4*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(768*g^3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^5*d^5*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*g^3*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (c*d*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)*(f + g*x)^5) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(5/2)/(6*g*(d + e*x)^(5/2)*(f + g*x)^6) + (5*c^6*d^6*ArcTan[(Sqrt[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])])/(512*g^(7/2)*(c*d*f - a*e*g)^(7/2))

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 876

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(d + e*x)^m*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^p/(g*(n + 1))), x] + Dist[c*(m/(e*g*(n + 1))), Int[(d +
e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{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] &&  !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p,
0] && LtQ[n, -1] &&  !(IntegerQ[n + p] && LeQ[n + p + 2, 0])

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 {\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)^7} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^6} \, dx}{12 g}\\ &=-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx}{8 g^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{64 g^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^4 d^4\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}{384 g^3 (c d f-a e g)}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^5 d^5\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}{512 g^3 (c d f-a e g)^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^6 d^6\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}{1024 g^3 (c d f-a e g)^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^6 d^6 e^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 )}{512 g^3 (c d f-a e g)^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \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 )}{512 g^{7/2} (c d f-a e g)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 2.99, size = 370, normalized size = 0.80 \begin {gather*} \frac {c^6 d^6 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (256 a^5 e^5 g^5+640 a^4 c d e^4 g^4 (-f+g x)+16 a^3 c^2 d^2 e^3 g^3 \left (27 f^2-106 f g x+27 g^2 x^2\right )+8 a^2 c^3 d^3 e^2 g^2 \left (-f^3+159 f^2 g x-159 f g^2 x^2+g^3 x^3\right )-2 a c^4 d^4 e g \left (5 f^4+28 f^3 g x-594 f^2 g^2 x^2+28 f g^3 x^3+5 g^4 x^4\right )+c^5 d^5 \left (-15 f^5-85 f^4 g x-198 f^3 g^2 x^2+198 f^2 g^3 x^3+85 f g^4 x^4+15 g^5 x^5\right )\right )}{c^6 d^6 (c d f-a e g)^3 (a e+c d x)^2 (f+g x)^6}+\frac {15 \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)^{7/2} (a e+c d x)^{5/2}}\right )}{1536 g^{7/2} (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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)^7),x]

[Out]

(c^6*d^6*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(256*a^5*e^5*g^5 + 640*a^4*c*d*e^4*g^4*(-f + g*x) + 16*a^3*
c^2*d^2*e^3*g^3*(27*f^2 - 106*f*g*x + 27*g^2*x^2) + 8*a^2*c^3*d^3*e^2*g^2*(-f^3 + 159*f^2*g*x - 159*f*g^2*x^2
+ g^3*x^3) - 2*a*c^4*d^4*e*g*(5*f^4 + 28*f^3*g*x - 594*f^2*g^2*x^2 + 28*f*g^3*x^3 + 5*g^4*x^4) + c^5*d^5*(-15*
f^5 - 85*f^4*g*x - 198*f^3*g^2*x^2 + 198*f^2*g^3*x^3 + 85*f*g^4*x^4 + 15*g^5*x^5)))/(c^6*d^6*(c*d*f - a*e*g)^3
*(a*e + c*d*x)^2*(f + g*x)^6) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e*g)^
(7/2)*(a*e + c*d*x)^(5/2))))/(1536*g^(7/2)*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1250\) vs. \(2(413)=826\).
time = 0.14, size = 1251, normalized size = 2.70

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-15 c^{5} d^{5} g^{5} x^{5} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f^{6}+90 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f \,g^{5} x^{5}+225 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f^{2} g^{4} x^{4}+10 a \,c^{4} d^{4} e \,g^{5} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+56 a \,c^{4} d^{4} e f \,g^{4} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+1272 a^{2} c^{3} d^{3} e^{2} f \,g^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-1188 a \,c^{4} d^{4} e \,f^{2} g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+1696 a^{3} c^{2} d^{2} e^{3} f \,g^{4} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-1272 a^{2} c^{3} d^{3} e^{2} f^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+56 a \,c^{4} d^{4} e \,f^{3} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-8 a^{2} c^{3} d^{3} e^{2} g^{5} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-432 a^{3} c^{2} d^{2} e^{3} g^{5} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-640 a^{4} c d \,e^{4} g^{5} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} g^{6} x^{6}+300 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f^{3} g^{3} x^{3}+225 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f^{4} g^{2} x^{2}+90 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{6} d^{6} f^{5} g x -256 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{5} e^{5} g^{5}+15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{5} d^{5} f^{5}-85 c^{5} d^{5} f \,g^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-198 c^{5} d^{5} f^{2} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+198 c^{5} d^{5} f^{3} g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+85 c^{5} d^{5} f^{4} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+640 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{4} c d \,e^{4} f \,g^{4}-432 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} c^{2} d^{2} e^{3} f^{2} g^{3}+8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c^{3} d^{3} e^{2} f^{3} g^{2}+10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{4} d^{4} e \,f^{4} g \right )}{1536 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{6} g^{3} \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d x +a e}}\)	\(1251\)

Verification of antiderivative is not currently implemented for this CAS.

[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)^7,x,method=_RETURNVERBOSE)

[Out]

1/1536*((c*d*x+a*e)*(e*x+d))^(1/2)*(-15*c^5*d^5*g^5*x^5*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+90*arctanh(g
*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f*g^5*x^5+225*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2))*c^6*d^6*f^2*g^4*x^4+300*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^3*g^3*x^3+225*
arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^4*g^2*x^2+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*
g-c*d*f)*g)^(1/2))*c^6*d^6*f^5*g*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^6+10*a*c^
4*d^4*e*g^5*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+56*a*c^4*d^4*e*f*g^4*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c
*d*f)*g)^(1/2)+1272*a^2*c^3*d^3*e^2*f*g^4*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-1188*a*c^4*d^4*e*f^2*g
^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+1696*a^3*c^2*d^2*e^3*f*g^4*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)
*g)^(1/2)-1272*a^2*c^3*d^3*e^2*f^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+56*a*c^4*d^4*e*f^3*g^2*x*(c
*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*a^2*c^3*d^3*e^2*g^5*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-43
2*a^3*c^2*d^2*e^3*g^5*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-640*a^4*c*d*e^4*g^5*x*(c*d*x+a*e)^(1/2)*((
a*e*g-c*d*f)*g)^(1/2)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*g^6*x^6-256*(c*d*x+a*e)^
(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^5*e^5*g^5+15*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^5*d^5*f^5-85*c^5*d^5*
f*g^4*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-198*c^5*d^5*f^2*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g
)^(1/2)+198*c^5*d^5*f^3*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+85*c^5*d^5*f^4*g*x*(c*d*x+a*e)^(1/2)
*((a*e*g-c*d*f)*g)^(1/2)+640*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^4*c*d*e^4*f*g^4-432*(c*d*x+a*e)^(1/2)
*((a*e*g-c*d*f)*g)^(1/2)*a^3*c^2*d^2*e^3*f^2*g^3+8*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*c^3*d^3*e^2*f
^3*g^2+10*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c^4*d^4*e*f^4*g)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(
g*x+f)^6/g^3/(a*e*g-c*d*f)/(a^2*e^2*g^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(c*d*x+a*e)^(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((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)^7,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1994 vs. \(2 (432) = 864\).
time = 23.52, size = 4027, normalized size = 8.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^7,x, algorithm="fricas")

[Out]

[1/3072*(15*(c^6*d^7*g^6*x^6 + 6*c^6*d^7*f*g^5*x^5 + 15*c^6*d^7*f^2*g^4*x^4 + 20*c^6*d^7*f^3*g^3*x^3 + 15*c^6*
d^7*f^4*g^2*x^2 + 6*c^6*d^7*f^5*g*x + c^6*d^7*f^6 + (c^6*d^6*g^6*x^7 + 6*c^6*d^6*f*g^5*x^6 + 15*c^6*d^6*f^2*g^
4*x^5 + 20*c^6*d^6*f^3*g^3*x^4 + 15*c^6*d^6*f^4*g^2*x^3 + 6*c^6*d^6*f^5*g*x^2 + c^6*d^6*f^6*x)*e)*sqrt(-c*d*f*
g + a*g^2*e)*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e + 2*sqrt(-c*d*f*g + a
*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*(15*c^
6*d^6*f*g^6*x^5 + 85*c^6*d^6*f^2*g^5*x^4 + 198*c^6*d^6*f^3*g^4*x^3 - 198*c^6*d^6*f^4*g^3*x^2 - 85*c^6*d^6*f^5*
g^2*x - 15*c^6*d^6*f^6*g - 256*a^6*g^7*e^6 - 128*(5*a^5*c*d*g^7*x - 7*a^5*c*d*f*g^6)*e^5 - 16*(27*a^4*c^2*d^2*
g^7*x^2 - 146*a^4*c^2*d^2*f*g^6*x + 67*a^4*c^2*d^2*f^2*g^5)*e^4 - 8*(a^3*c^3*d^3*g^7*x^3 - 213*a^3*c^3*d^3*f*g
^6*x^2 + 371*a^3*c^3*d^3*f^2*g^5*x - 55*a^3*c^3*d^3*f^3*g^4)*e^3 + 2*(5*a^2*c^4*d^4*g^7*x^4 + 32*a^2*c^4*d^4*f
*g^6*x^3 - 1230*a^2*c^4*d^4*f^2*g^5*x^2 + 664*a^2*c^4*d^4*f^3*g^4*x + a^2*c^4*d^4*f^4*g^3)*e^2 - (15*a*c^5*d^5
*g^7*x^5 + 95*a*c^5*d^5*f*g^6*x^4 + 254*a*c^5*d^5*f^2*g^5*x^3 - 1386*a*c^5*d^5*f^3*g^4*x^2 - 29*a*c^5*d^5*f^4*
g^3*x - 5*a*c^5*d^5*f^5*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*f^4*g^10*x
^6 + 6*c^4*d^5*f^5*g^9*x^5 + 15*c^4*d^5*f^6*g^8*x^4 + 20*c^4*d^5*f^7*g^7*x^3 + 15*c^4*d^5*f^8*g^6*x^2 + 6*c^4*
d^5*f^9*g^5*x + c^4*d^5*f^10*g^4 + (a^4*g^14*x^7 + 6*a^4*f*g^13*x^6 + 15*a^4*f^2*g^12*x^5 + 20*a^4*f^3*g^11*x^
4 + 15*a^4*f^4*g^10*x^3 + 6*a^4*f^5*g^9*x^2 + a^4*f^6*g^8*x)*e^5 - (4*a^3*c*d*f*g^13*x^7 - a^4*d*f^6*g^8 + (24
*a^3*c*d*f^2*g^12 - a^4*d*g^14)*x^6 + 6*(10*a^3*c*d*f^3*g^11 - a^4*d*f*g^13)*x^5 + 5*(16*a^3*c*d*f^4*g^10 - 3*
a^4*d*f^2*g^12)*x^4 + 20*(3*a^3*c*d*f^5*g^9 - a^4*d*f^3*g^11)*x^3 + 3*(8*a^3*c*d*f^6*g^8 - 5*a^4*d*f^4*g^10)*x
^2 + 2*(2*a^3*c*d*f^7*g^7 - 3*a^4*d*f^5*g^9)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^12*x^7 - 2*a^3*c*d^2*f^7*g^7 + 2*
(9*a^2*c^2*d^2*f^3*g^11 - a^3*c*d^2*f*g^13)*x^6 + 3*(15*a^2*c^2*d^2*f^4*g^10 - 4*a^3*c*d^2*f^2*g^12)*x^5 + 30*
(2*a^2*c^2*d^2*f^5*g^9 - a^3*c*d^2*f^3*g^11)*x^4 + 5*(9*a^2*c^2*d^2*f^6*g^8 - 8*a^3*c*d^2*f^4*g^10)*x^3 + 6*(3
*a^2*c^2*d^2*f^7*g^7 - 5*a^3*c*d^2*f^5*g^9)*x^2 + 3*(a^2*c^2*d^2*f^8*g^6 - 4*a^3*c*d^2*f^6*g^8)*x)*e^3 - 2*(2*
a*c^3*d^3*f^3*g^11*x^7 - 3*a^2*c^2*d^3*f^8*g^6 + 3*(4*a*c^3*d^3*f^4*g^10 - a^2*c^2*d^3*f^2*g^12)*x^6 + 6*(5*a*
c^3*d^3*f^5*g^9 - 3*a^2*c^2*d^3*f^3*g^11)*x^5 + 5*(8*a*c^3*d^3*f^6*g^8 - 9*a^2*c^2*d^3*f^4*g^10)*x^4 + 30*(a*c
^3*d^3*f^7*g^7 - 2*a^2*c^2*d^3*f^5*g^9)*x^3 + 3*(4*a*c^3*d^3*f^8*g^6 - 15*a^2*c^2*d^3*f^6*g^8)*x^2 + 2*(a*c^3*
d^3*f^9*g^5 - 9*a^2*c^2*d^3*f^7*g^7)*x)*e^2 + (c^4*d^4*f^4*g^10*x^7 - 4*a*c^3*d^4*f^9*g^5 + 2*(3*c^4*d^4*f^5*g
^9 - 2*a*c^3*d^4*f^3*g^11)*x^6 + 3*(5*c^4*d^4*f^6*g^8 - 8*a*c^3*d^4*f^4*g^10)*x^5 + 20*(c^4*d^4*f^7*g^7 - 3*a*
c^3*d^4*f^5*g^9)*x^4 + 5*(3*c^4*d^4*f^8*g^6 - 16*a*c^3*d^4*f^6*g^8)*x^3 + 6*(c^4*d^4*f^9*g^5 - 10*a*c^3*d^4*f^
7*g^7)*x^2 + (c^4*d^4*f^10*g^4 - 24*a*c^3*d^4*f^8*g^6)*x)*e), -1/1536*(15*(c^6*d^7*g^6*x^6 + 6*c^6*d^7*f*g^5*x
^5 + 15*c^6*d^7*f^2*g^4*x^4 + 20*c^6*d^7*f^3*g^3*x^3 + 15*c^6*d^7*f^4*g^2*x^2 + 6*c^6*d^7*f^5*g*x + c^6*d^7*f^
6 + (c^6*d^6*g^6*x^7 + 6*c^6*d^6*f*g^5*x^6 + 15*c^6*d^6*f^2*g^4*x^5 + 20*c^6*d^6*f^3*g^3*x^4 + 15*c^6*d^6*f^4*
g^2*x^3 + 6*c^6*d^6*f^5*g*x^2 + c^6*d^6*f^6*x)*e)*sqrt(c*d*f*g - a*g^2*e)*arctan(sqrt(c*d*f*g - a*g^2*e)*sqrt(
c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) - (15*c^
6*d^6*f*g^6*x^5 + 85*c^6*d^6*f^2*g^5*x^4 + 198*c^6*d^6*f^3*g^4*x^3 - 198*c^6*d^6*f^4*g^3*x^2 - 85*c^6*d^6*f^5*
g^2*x - 15*c^6*d^6*f^6*g - 256*a^6*g^7*e^6 - 128*(5*a^5*c*d*g^7*x - 7*a^5*c*d*f*g^6)*e^5 - 16*(27*a^4*c^2*d^2*
g^7*x^2 - 146*a^4*c^2*d^2*f*g^6*x + 67*a^4*c^2*d^2*f^2*g^5)*e^4 - 8*(a^3*c^3*d^3*g^7*x^3 - 213*a^3*c^3*d^3*f*g
^6*x^2 + 371*a^3*c^3*d^3*f^2*g^5*x - 55*a^3*c^3*d^3*f^3*g^4)*e^3 + 2*(5*a^2*c^4*d^4*g^7*x^4 + 32*a^2*c^4*d^4*f
*g^6*x^3 - 1230*a^2*c^4*d^4*f^2*g^5*x^2 + 664*a^2*c^4*d^4*f^3*g^4*x + a^2*c^4*d^4*f^4*g^3)*e^2 - (15*a*c^5*d^5
*g^7*x^5 + 95*a*c^5*d^5*f*g^6*x^4 + 254*a*c^5*d^5*f^2*g^5*x^3 - 1386*a*c^5*d^5*f^3*g^4*x^2 - 29*a*c^5*d^5*f^4*
g^3*x - 5*a*c^5*d^5*f^5*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*f^4*g^10*x
^6 + 6*c^4*d^5*f^5*g^9*x^5 + 15*c^4*d^5*f^6*g^8*x^4 + 20*c^4*d^5*f^7*g^7*x^3 + 15*c^4*d^5*f^8*g^6*x^2 + 6*c^4*
d^5*f^9*g^5*x + c^4*d^5*f^10*g^4 + (a^4*g^14*x^7 + 6*a^4*f*g^13*x^6 + 15*a^4*f^2*g^12*x^5 + 20*a^4*f^3*g^11*x^
4 + 15*a^4*f^4*g^10*x^3 + 6*a^4*f^5*g^9*x^2 + a^4*f^6*g^8*x)*e^5 - (4*a^3*c*d*f*g^13*x^7 - a^4*d*f^6*g^8 + (24
*a^3*c*d*f^2*g^12 - a^4*d*g^14)*x^6 + 6*(10*a^3*c*d*f^3*g^11 - a^4*d*f*g^13)*x^5 + 5*(16*a^3*c*d*f^4*g^10 - 3*
a^4*d*f^2*g^12)*x^4 + 20*(3*a^3*c*d*f^5*g^9 - a^4*d*f^3*g^11)*x^3 + 3*(8*a^3*c*d*f^6*g^8 - 5*a^4*d*f^4*g^10)*x
^2 + 2*(2*a^3*c*d*f^7*g^7 - 3*a^4*d*f^5*g^9)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^12*x^7 - 2*a^3*c*d^2*f^7*g^7 + 2*
(9*a^2*c^2*d^2*f^3*g^11 - a^3*c*d^2*f*g^13)*x^6...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)**7,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5987 deep

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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((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)^7,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 (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^7\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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