3.3.95 \(\int (g+h x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [295]

Optimal. Leaf size=158 \[ -\frac {B (b c-a d) h (3 b d g-b c h-a d h) n x}{3 b^2 d^2}-\frac {B (b c-a d) h^2 n x^2}{6 b d}-\frac {B (b g-a h)^3 n \log (a+b x)}{3 b^3 h}+\frac {B (d g-c h)^3 n \log (c+d x)}{3 d^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 h} \]

[Out]

-1/3*B*(-a*d+b*c)*h*(-a*d*h-b*c*h+3*b*d*g)*n*x/b^2/d^2-1/6*B*(-a*d+b*c)*h^2*n*x^2/b/d-1/3*B*(-a*h+b*g)^3*n*ln(
b*x+a)/b^3/h+1/3*B*(-c*h+d*g)^3*n*ln(d*x+c)/d^3/h+1/3*(h*x+g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/h

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Rubi [A]
time = 0.11, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 84} \begin {gather*} \frac {(g+h x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 h}-\frac {B n (b g-a h)^3 \log (a+b x)}{3 b^3 h}-\frac {B h n x (b c-a d) (-a d h-b c h+3 b d g)}{3 b^2 d^2}-\frac {B h^2 n x^2 (b c-a d)}{6 b d}+\frac {B n (d g-c h)^3 \log (c+d x)}{3 d^3 h} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

-1/3*(B*(b*c - a*d)*h*(3*b*d*g - b*c*h - a*d*h)*n*x)/(b^2*d^2) - (B*(b*c - a*d)*h^2*n*x^2)/(6*b*d) - (B*(b*g -
a*h)^3*n*Log[a + b*x])/(3*b^3*h) + (B*(d*g - c*h)^3*n*Log[c + d*x])/(3*d^3*h) + ((g + h*x)^3*(A + B*Log[(e*(a
+ b*x)^n)/(c + d*x)^n]))/(3*h)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps

\begin {align*} \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (g+h x)^2+B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A (g+h x)^3}{3 h}+B \int (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A (g+h x)^3}{3 h}+\frac {B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {(B (b c-a d) n) \int \frac {(g+h x)^3}{(a+b x) (c+d x)} \, dx}{3 h}\\ &=\frac {A (g+h x)^3}{3 h}+\frac {B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {(B (b c-a d) n) \int \left (\frac {h^2 (3 b d g-b c h-a d h)}{b^2 d^2}+\frac {h^3 x}{b d}+\frac {(b g-a h)^3}{b^2 (b c-a d) (a+b x)}+\frac {(d g-c h)^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{3 h}\\ &=-\frac {B (b c-a d) h (3 b d g-b c h-a d h) n x}{3 b^2 d^2}-\frac {B (b c-a d) h^2 n x^2}{6 b d}+\frac {A (g+h x)^3}{3 h}-\frac {B (b g-a h)^3 n \log (a+b x)}{3 b^3 h}+\frac {B (d g-c h)^3 n \log (c+d x)}{3 d^3 h}+\frac {B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 195, normalized size = 1.23 \begin {gather*} \frac {2 a B d^3 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) n \log (a+b x)-2 b^3 B c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) n \log (c+d x)+b d x \left (B (b c-a d) h n (-6 b d g+2 b c h+2 a d h-b d h x)+2 A b^2 d^2 \left (3 g^2+3 g h x+h^2 x^2\right )+2 b^2 B d^2 \left (3 g^2+3 g h x+h^2 x^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{6 b^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(2*a*B*d^3*(3*b^2*g^2 - 3*a*b*g*h + a^2*h^2)*n*Log[a + b*x] - 2*b^3*B*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2)*n*Lo
g[c + d*x] + b*d*x*(B*(b*c - a*d)*h*n*(-6*b*d*g + 2*b*c*h + 2*a*d*h - b*d*h*x) + 2*A*b^2*d^2*(3*g^2 + 3*g*h*x
+ h^2*x^2) + 2*b^2*B*d^2*(3*g^2 + 3*g*h*x + h^2*x^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(6*b^3*d^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.38, size = 1425, normalized size = 9.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(1425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x,method=_RETURNVERBOSE)

[Out]

1/3*h^2*B*x^3*ln((b*x+a)^n)+1/3*h^2*B*ln(e)*x^3+1/3/h*B*g^3*ln((b*x+a)^n)+B*g^2*x*ln((b*x+a)^n)+B*ln(e)*g^2*x+
1/3*A*h^2*x^3+x*A*g^2+1/b*h*B*a*g*n*x-h/d*B*c*g*n*x+1/2*I*h*B*Pi*g*x^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*
x+c)^n))^2-1/6*I*h^2*B*Pi*x^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-1/6*I*h^2*B*
Pi*x^3*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+1/2*I*h*B*Pi*g*x^2*csgn(I/((d*x
+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*h*B*Pi*g*x^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*h
*B*Pi*g*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*I*B*Pi*g^2*x*csgn(I*(b*x+a)^n)
*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-1/2*I*B*Pi*g^2*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*cs
gn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*h*B*Pi*g*x^2*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+
c)^n))-1/2*I*h*B*Pi*g*x^2*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+h*A*g*x^2-1/
d*B*ln(d*x+c)*c*g^2*n+1/b*B*ln(-b*x-a)*a*g^2*n-1/3*h^2/d^3*B*ln(d*x+c)*c^3*n+1/3/b^3*h^2*B*ln(-b*x-a)*a^3*n-1/
2*I*B*Pi*g^2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/6*I*h^2*B*Pi*x^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/6*I*h^2*
B*Pi*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/2*I*B*Pi*g^2*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/3*(h*x+g)^3*B/h*
ln((d*x+c)^n)+h*B*g*x^2*ln((b*x+a)^n)+h*B*ln(e)*g*x^2+1/3/h*B*ln(d*x+c)*g^3*n-1/3/h*B*ln(-b*x-a)*g^3*n+1/6/b*h
^2*B*a*n*x^2-1/6*h^2/d*B*c*n*x^2-1/3/b^2*h^2*B*a^2*n*x+1/3*h^2/d^2*B*c^2*n*x+h/d^2*B*ln(d*x+c)*c^2*g*n-1/b^2*h
*B*ln(-b*x-a)*a^2*g*n+1/2*I*B*Pi*g^2*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*B*Pi*g^2*x*cs
gn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*g^2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*
(b*x+a)^n)^2+1/2*I*B*Pi*g^2*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/6*I*h^2*B*Pi*x^3*csgn(I/((d*
x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/6*I*h^2*B*Pi*x^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/6*I*
h^2*B*Pi*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/6*I*h^2*B*Pi*x^3*csgn(I*(b*x+a)
^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1/2*I*h*B*Pi*g*x^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/2*I*h*B*Pi*g*x^2*cs
gn(I*(b*x+a)^n/((d*x+c)^n))^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (149) = 298\).
time = 0.29, size = 300, normalized size = 1.90 \begin {gather*} \frac {1}{3} \, B h^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{3} \, A h^{2} x^{3} + B g h x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g h x^{2} + {\left (\frac {a n e \log \left (b x + a\right )}{b} - \frac {c n e \log \left (d x + c\right )}{d}\right )} B g^{2} e^{\left (-1\right )} - {\left (\frac {a^{2} n e \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} n e \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c n - a d n\right )} x e}{b d}\right )} B g h e^{\left (-1\right )} + \frac {1}{6} \, {\left (\frac {2 \, a^{3} n e \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} n e \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d n - a b d^{2} n\right )} x^{2} e - 2 \, {\left (b^{2} c^{2} n - a^{2} d^{2} n\right )} x e}{b^{2} d^{2}}\right )} B h^{2} e^{\left (-1\right )} + B g^{2} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A g^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

1/3*B*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A*h^2*x^3 + B*g*h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + A*g*
h*x^2 + (a*n*e*log(b*x + a)/b - c*n*e*log(d*x + c)/d)*B*g^2*e^(-1) - (a^2*n*e*log(b*x + a)/b^2 - c^2*n*e*log(d
*x + c)/d^2 + (b*c*n - a*d*n)*x*e/(b*d))*B*g*h*e^(-1) + 1/6*(2*a^3*n*e*log(b*x + a)/b^3 - 2*c^3*n*e*log(d*x +
c)/d^3 - ((b^2*c*d*n - a*b*d^2*n)*x^2*e - 2*(b^2*c^2*n - a^2*d^2*n)*x*e)/(b^2*d^2))*B*h^2*e^(-1) + B*g^2*x*log
((b*x + a)^n*e/(d*x + c)^n) + A*g^2*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (149) = 298\).
time = 0.37, size = 325, normalized size = 2.06 \begin {gather*} \frac {2 \, {\left (A + B\right )} b^{3} d^{3} h^{2} x^{3} + {\left (6 \, {\left (A + B\right )} b^{3} d^{3} g h - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} h^{2} n\right )} x^{2} + 2 \, {\left (3 \, {\left (A + B\right )} b^{3} d^{3} g^{2} - {\left (3 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g h - {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} h^{2}\right )} n\right )} x + 2 \, {\left (B b^{3} d^{3} h^{2} n x^{3} + 3 \, B b^{3} d^{3} g h n x^{2} + 3 \, B b^{3} d^{3} g^{2} n x + {\left (3 \, B a b^{2} d^{3} g^{2} - 3 \, B a^{2} b d^{3} g h + B a^{3} d^{3} h^{2}\right )} n\right )} \log \left (b x + a\right ) - 2 \, {\left (B b^{3} d^{3} h^{2} n x^{3} + 3 \, B b^{3} d^{3} g h n x^{2} + 3 \, B b^{3} d^{3} g^{2} n x + {\left (3 \, B b^{3} c d^{2} g^{2} - 3 \, B b^{3} c^{2} d g h + B b^{3} c^{3} h^{2}\right )} n\right )} \log \left (d x + c\right )}{6 \, b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

1/6*(2*(A + B)*b^3*d^3*h^2*x^3 + (6*(A + B)*b^3*d^3*g*h - (B*b^3*c*d^2 - B*a*b^2*d^3)*h^2*n)*x^2 + 2*(3*(A + B
)*b^3*d^3*g^2 - (3*(B*b^3*c*d^2 - B*a*b^2*d^3)*g*h - (B*b^3*c^2*d - B*a^2*b*d^3)*h^2)*n)*x + 2*(B*b^3*d^3*h^2*
n*x^3 + 3*B*b^3*d^3*g*h*n*x^2 + 3*B*b^3*d^3*g^2*n*x + (3*B*a*b^2*d^3*g^2 - 3*B*a^2*b*d^3*g*h + B*a^3*d^3*h^2)*
n)*log(b*x + a) - 2*(B*b^3*d^3*h^2*n*x^3 + 3*B*b^3*d^3*g*h*n*x^2 + 3*B*b^3*d^3*g^2*n*x + (3*B*b^3*c*d^2*g^2 -
3*B*b^3*c^2*d*g*h + B*b^3*c^3*h^2)*n)*log(d*x + c))/(b^3*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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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((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.47, size = 372, normalized size = 2.35 \begin {gather*} x^2\,\left (\frac {3\,A\,a\,d\,h^2+3\,A\,b\,c\,h^2+6\,A\,b\,d\,g\,h+B\,a\,d\,h^2\,n-B\,b\,c\,h^2\,n}{6\,b\,d}-\frac {A\,h^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,b\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (B\,g^2\,x+B\,g\,h\,x^2+\frac {B\,h^2\,x^3}{3}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {3\,A\,a\,d\,h^2+3\,A\,b\,c\,h^2+6\,A\,b\,d\,g\,h+B\,a\,d\,h^2\,n-B\,b\,c\,h^2\,n}{3\,b\,d}-\frac {A\,h^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}\right )}{3\,b\,d}-\frac {3\,A\,a\,c\,h^2+3\,A\,b\,d\,g^2+6\,A\,a\,d\,g\,h+6\,A\,b\,c\,g\,h+3\,B\,a\,d\,g\,h\,n-3\,B\,b\,c\,g\,h\,n}{3\,b\,d}+\frac {A\,a\,c\,h^2}{b\,d}\right )+\frac {A\,h^2\,x^3}{3}+\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^3\,h^2-3\,B\,n\,a^2\,b\,g\,h+3\,B\,n\,a\,b^2\,g^2\right )}{3\,b^3}-\frac {\ln \left (c+d\,x\right )\,\left (B\,n\,c^3\,h^2-3\,B\,n\,c^2\,d\,g\,h+3\,B\,n\,c\,d^2\,g^2\right )}{3\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)^2*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n)),x)

[Out]

x^2*((3*A*a*d*h^2 + 3*A*b*c*h^2 + 6*A*b*d*g*h + B*a*d*h^2*n - B*b*c*h^2*n)/(6*b*d) - (A*h^2*(3*a*d + 3*b*c))/(
6*b*d)) + log((e*(a + b*x)^n)/(c + d*x)^n)*((B*h^2*x^3)/3 + B*g^2*x + B*g*h*x^2) - x*(((3*a*d + 3*b*c)*((3*A*a
*d*h^2 + 3*A*b*c*h^2 + 6*A*b*d*g*h + B*a*d*h^2*n - B*b*c*h^2*n)/(3*b*d) - (A*h^2*(3*a*d + 3*b*c))/(3*b*d)))/(3
*b*d) - (3*A*a*c*h^2 + 3*A*b*d*g^2 + 6*A*a*d*g*h + 6*A*b*c*g*h + 3*B*a*d*g*h*n - 3*B*b*c*g*h*n)/(3*b*d) + (A*a
*c*h^2)/(b*d)) + (A*h^2*x^3)/3 + (log(a + b*x)*(B*a^3*h^2*n + 3*B*a*b^2*g^2*n - 3*B*a^2*b*g*h*n))/(3*b^3) - (l
og(c + d*x)*(B*c^3*h^2*n + 3*B*c*d^2*g^2*n - 3*B*c^2*d*g*h*n))/(3*d^3)

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