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

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

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*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))

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Rubi [A]  time = 0.716461, antiderivative size = 463, normalized size of antiderivative = 1., 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 = {862, 872, 874, 205} \[ \frac{5 c^5 d^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{768 g^3 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 g^3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt{d+e x} (f+g x)^4}+\frac{5 c^6 d^6 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6} \]

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]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*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 862

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 872

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 874

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]

Rule 205

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

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 ) \operatorname{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*}

Mathematica [C]  time = 0.103569, size = 79, normalized size = 0.17 \[ \frac{2 c^6 d^6 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (\frac{7}{2},7;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{7 (d+e x)^{7/2} (c d f-a e g)^7} \]

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]

(2*c^6*d^6*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 7, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)
])/(7*(c*d*f - a*e*g)^7*(d + e*x)^(7/2))

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Maple [B]  time = 0.351, size = 1261, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

1/1536*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(56*x^3*a*c^4*d^4*e*f*g^4*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(
1/2)+1272*x^2*a^2*c^3*d^3*e^2*f*g^4*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-1188*x^2*a*c^4*d^4*e*f^2*g^3*((a
*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+56*x*a*c^4*d^4*e*f^3*g^2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+1696
*x*a^3*c^2*d^2*e^3*f*g^4*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*
f)*g)^(1/2))*x^6*c^6*d^6*g^6+15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^6-256*((a*e*g-c
*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^5*e^5*g^5+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^5*d^5*f^5+300*arct
anh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^3*c^6*d^6*f^3*g^3-432*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(
1/2)*a^3*c^2*d^2*e^3*f^2*g^3-85*x^4*c^5*d^5*f*g^4*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-198*x^3*c^5*d^5*f^
2*g^3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+198*x^2*c^5*d^5*f^3*g^2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1
/2)+85*x*c^5*d^5*f^4*g*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-1272*x*a^2*c^3*d^3*e^2*f^2*g^3*((a*e*g-c*d*f)
*g)^(1/2)*(c*d*x+a*e)^(1/2)+225*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^2*c^6*d^6*f^4*g^2+90*ar
ctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*c^6*d^6*f^5*g+90*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*
f)*g)^(1/2))*x^5*c^6*d^6*f*g^5+225*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^4*c^6*d^6*f^2*g^4-15
*x^5*c^5*d^5*g^5*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-432*x^2*a^3*c^2*d^2*e^3*g^5*((a*e*g-c*d*f)*g)^(1/2)
*(c*d*x+a*e)^(1/2)+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^3*d^3*e^2*f^3*g^2+10*((a*e*g-c*d*f)*g)^(1
/2)*(c*d*x+a*e)^(1/2)*a*c^4*d^4*e*f^4*g+640*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^4*c*d*e^4*f*g^4-640*x*
a^4*c*d*e^4*g^5*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+10*x^4*a*c^4*d^4*e*g^5*((a*e*g-c*d*f)*g)^(1/2)*(c*d*
x+a*e)^(1/2)-8*x^3*a^2*c^3*d^3*e^2*g^5*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2))/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{7}}\,{d x} \end{align*}

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

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Fricas [B]  time = 2.39786, size = 7769, normalized size = 16.78 \begin{align*} \text{result too large to display} \end{align*}

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^6*e*g^6*x^7 + c^6*d^7*f^6 + (6*c^6*d^6*e*f*g^5 + c^6*d^7*g^6)*x^6 + 3*(5*c^6*d^6*e*f^2*g^4
+ 2*c^6*d^7*f*g^5)*x^5 + 5*(4*c^6*d^6*e*f^3*g^3 + 3*c^6*d^7*f^2*g^4)*x^4 + 5*(3*c^6*d^6*e*f^4*g^2 + 4*c^6*d^7*
f^3*g^3)*x^3 + 3*(2*c^6*d^6*e*f^5*g + 5*c^6*d^7*f^4*g^2)*x^2 + (c^6*d^6*e*f^6 + 6*c^6*d^7*f^5*g)*x)*sqrt(-c*d*
f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x + 2*sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*(15*
c^6*d^6*f^6*g - 5*a*c^5*d^5*e*f^5*g^2 - 2*a^2*c^4*d^4*e^2*f^4*g^3 - 440*a^3*c^3*d^3*e^3*f^3*g^4 + 1072*a^4*c^2
*d^2*e^4*f^2*g^5 - 896*a^5*c*d*e^5*f*g^6 + 256*a^6*e^6*g^7 - 15*(c^6*d^6*f*g^6 - a*c^5*d^5*e*g^7)*x^5 - 5*(17*
c^6*d^6*f^2*g^5 - 19*a*c^5*d^5*e*f*g^6 + 2*a^2*c^4*d^4*e^2*g^7)*x^4 - 2*(99*c^6*d^6*f^3*g^4 - 127*a*c^5*d^5*e*
f^2*g^5 + 32*a^2*c^4*d^4*e^2*f*g^6 - 4*a^3*c^3*d^3*e^3*g^7)*x^3 + 6*(33*c^6*d^6*f^4*g^3 - 231*a*c^5*d^5*e*f^3*
g^4 + 410*a^2*c^4*d^4*e^2*f^2*g^5 - 284*a^3*c^3*d^3*e^3*f*g^6 + 72*a^4*c^2*d^2*e^4*g^7)*x^2 + (85*c^6*d^6*f^5*
g^2 - 29*a*c^5*d^5*e*f^4*g^3 - 1328*a^2*c^4*d^4*e^2*f^3*g^4 + 2968*a^3*c^3*d^3*e^3*f^2*g^5 - 2336*a^4*c^2*d^2*
e^4*f*g^6 + 640*a^5*c*d*e^5*g^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^10*g
^4 - 4*a*c^3*d^4*e*f^9*g^5 + 6*a^2*c^2*d^3*e^2*f^8*g^6 - 4*a^3*c*d^2*e^3*f^7*g^7 + a^4*d*e^4*f^6*g^8 + (c^4*d^
4*e*f^4*g^10 - 4*a*c^3*d^3*e^2*f^3*g^11 + 6*a^2*c^2*d^2*e^3*f^2*g^12 - 4*a^3*c*d*e^4*f*g^13 + a^4*e^5*g^14)*x^
7 + (6*c^4*d^4*e*f^5*g^9 + a^4*d*e^4*g^14 + (c^4*d^5 - 24*a*c^3*d^3*e^2)*f^4*g^10 - 4*(a*c^3*d^4*e - 9*a^2*c^2
*d^2*e^3)*f^3*g^11 + 6*(a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^2*g^12 - 2*(2*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^13)*x^
6 + 3*(5*c^4*d^4*e*f^6*g^8 + 2*a^4*d*e^4*f*g^13 + 2*(c^4*d^5 - 10*a*c^3*d^3*e^2)*f^5*g^9 - 2*(4*a*c^3*d^4*e -
15*a^2*c^2*d^2*e^3)*f^4*g^10 + 4*(3*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*f^3*g^11 - (8*a^3*c*d^2*e^3 - 5*a^4*e^5)*
f^2*g^12)*x^5 + 5*(4*c^4*d^4*e*f^7*g^7 + 3*a^4*d*e^4*f^2*g^12 + (3*c^4*d^5 - 16*a*c^3*d^3*e^2)*f^6*g^8 - 12*(a
*c^3*d^4*e - 2*a^2*c^2*d^2*e^3)*f^5*g^9 + 2*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^4*g^10 - 4*(3*a^3*c*d^2*e^3
- a^4*e^5)*f^3*g^11)*x^4 + 5*(3*c^4*d^4*e*f^8*g^6 + 4*a^4*d*e^4*f^3*g^11 + 4*(c^4*d^5 - 3*a*c^3*d^3*e^2)*f^7*g
^7 - 2*(8*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^6*g^8 + 12*(2*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^9 - (16*a^3*c*
d^2*e^3 - 3*a^4*e^5)*f^4*g^10)*x^3 + 3*(2*c^4*d^4*e*f^9*g^5 + 5*a^4*d*e^4*f^4*g^10 + (5*c^4*d^5 - 8*a*c^3*d^3*
e^2)*f^8*g^6 - 4*(5*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^7*g^7 + 2*(15*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^6*g^8
- 2*(10*a^3*c*d^2*e^3 - a^4*e^5)*f^5*g^9)*x^2 + (c^4*d^4*e*f^10*g^4 + 6*a^4*d*e^4*f^5*g^9 + 2*(3*c^4*d^5 - 2*a
*c^3*d^3*e^2)*f^9*g^5 - 6*(4*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^8*g^6 + 4*(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^7*
g^7 - (24*a^3*c*d^2*e^3 - a^4*e^5)*f^6*g^8)*x), -1/1536*(15*(c^6*d^6*e*g^6*x^7 + c^6*d^7*f^6 + (6*c^6*d^6*e*f*
g^5 + c^6*d^7*g^6)*x^6 + 3*(5*c^6*d^6*e*f^2*g^4 + 2*c^6*d^7*f*g^5)*x^5 + 5*(4*c^6*d^6*e*f^3*g^3 + 3*c^6*d^7*f^
2*g^4)*x^4 + 5*(3*c^6*d^6*e*f^4*g^2 + 4*c^6*d^7*f^3*g^3)*x^3 + 3*(2*c^6*d^6*e*f^5*g + 5*c^6*d^7*f^4*g^2)*x^2 +
(c^6*d^6*e*f^6 + 6*c^6*d^7*f^5*g)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) + (15*c^6*d^6*f^6*g -
5*a*c^5*d^5*e*f^5*g^2 - 2*a^2*c^4*d^4*e^2*f^4*g^3 - 440*a^3*c^3*d^3*e^3*f^3*g^4 + 1072*a^4*c^2*d^2*e^4*f^2*g^5
- 896*a^5*c*d*e^5*f*g^6 + 256*a^6*e^6*g^7 - 15*(c^6*d^6*f*g^6 - a*c^5*d^5*e*g^7)*x^5 - 5*(17*c^6*d^6*f^2*g^5
- 19*a*c^5*d^5*e*f*g^6 + 2*a^2*c^4*d^4*e^2*g^7)*x^4 - 2*(99*c^6*d^6*f^3*g^4 - 127*a*c^5*d^5*e*f^2*g^5 + 32*a^2
*c^4*d^4*e^2*f*g^6 - 4*a^3*c^3*d^3*e^3*g^7)*x^3 + 6*(33*c^6*d^6*f^4*g^3 - 231*a*c^5*d^5*e*f^3*g^4 + 410*a^2*c^
4*d^4*e^2*f^2*g^5 - 284*a^3*c^3*d^3*e^3*f*g^6 + 72*a^4*c^2*d^2*e^4*g^7)*x^2 + (85*c^6*d^6*f^5*g^2 - 29*a*c^5*d
^5*e*f^4*g^3 - 1328*a^2*c^4*d^4*e^2*f^3*g^4 + 2968*a^3*c^3*d^3*e^3*f^2*g^5 - 2336*a^4*c^2*d^2*e^4*f*g^6 + 640*
a^5*c*d*e^5*g^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^10*g^4 - 4*a*c^3*d^4
*e*f^9*g^5 + 6*a^2*c^2*d^3*e^2*f^8*g^6 - 4*a^3*c*d^2*e^3*f^7*g^7 + a^4*d*e^4*f^6*g^8 + (c^4*d^4*e*f^4*g^10 - 4
*a*c^3*d^3*e^2*f^3*g^11 + 6*a^2*c^2*d^2*e^3*f^2*g^12 - 4*a^3*c*d*e^4*f*g^13 + a^4*e^5*g^14)*x^7 + (6*c^4*d^4*e
*f^5*g^9 + a^4*d*e^4*g^14 + (c^4*d^5 - 24*a*c^3*d^3*e^2)*f^4*g^10 - 4*(a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^
11 + 6*(a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^2*g^12 - 2*(2*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^13)*x^6 + 3*(5*c^4*d^4
*e*f^6*g^8 + 2*a^4*d*e^4*f*g^13 + 2*(c^4*d^5 - 10*a*c^3*d^3*e^2)*f^5*g^9 - 2*(4*a*c^3*d^4*e - 15*a^2*c^2*d^2*e
^3)*f^4*g^10 + 4*(3*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*f^3*g^11 - (8*a^3*c*d^2*e^3 - 5*a^4*e^5)*f^2*g^12)*x^5 +
5*(4*c^4*d^4*e*f^7*g^7 + 3*a^4*d*e^4*f^2*g^12 + (3*c^4*d^5 - 16*a*c^3*d^3*e^2)*f^6*g^8 - 12*(a*c^3*d^4*e - 2*a
^2*c^2*d^2*e^3)*f^5*g^9 + 2*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^4*g^10 - 4*(3*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g
^11)*x^4 + 5*(3*c^4*d^4*e*f^8*g^6 + 4*a^4*d*e^4*f^3*g^11 + 4*(c^4*d^5 - 3*a*c^3*d^3*e^2)*f^7*g^7 - 2*(8*a*c^3*
d^4*e - 9*a^2*c^2*d^2*e^3)*f^6*g^8 + 12*(2*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^9 - (16*a^3*c*d^2*e^3 - 3*a^4*
e^5)*f^4*g^10)*x^3 + 3*(2*c^4*d^4*e*f^9*g^5 + 5*a^4*d*e^4*f^4*g^10 + (5*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^8*g^6 - 4
*(5*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^7*g^7 + 2*(15*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^6*g^8 - 2*(10*a^3*c*d^
2*e^3 - a^4*e^5)*f^5*g^9)*x^2 + (c^4*d^4*e*f^10*g^4 + 6*a^4*d*e^4*f^5*g^9 + 2*(3*c^4*d^5 - 2*a*c^3*d^3*e^2)*f^
9*g^5 - 6*(4*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^8*g^6 + 4*(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^7*g^7 - (24*a^3*c*
d^2*e^3 - a^4*e^5)*f^6*g^8)*x)]

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

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]

Timed out

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

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