3.3124 \(\int (a+b x)^m (c+d x)^n \left (\frac{b c f+a d f+a d f m+b c f n}{b d (2+m+n)}+f x\right )^{-3-m-n} \, dx\)

Optimal. Leaf size=88 \[ \frac{b d (m+n+2) (a+b x)^{m+1} (c+d x)^{n+1} \left (\frac{f (a d (m+1)+b c (n+1))}{b d (m+n+2)}+f x\right )^{-m-n-2}}{f (m+1) (n+1) (b c-a d)^2} \]

[Out]

(b*d*(2 + m + n)*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*((f*(a*d*(1 + m) + b*c*(1 +
n)))/(b*d*(2 + m + n)) + f*x)^(-2 - m - n))/((b*c - a*d)^2*f*(1 + m)*(1 + n))

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Rubi [A]  time = 0.227201, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.017 \[ \frac{b d (m+n+2) (a+b x)^{m+1} (c+d x)^{n+1} \left (\frac{f (a d (m+1)+b c (n+1))}{b d (m+n+2)}+f x\right )^{-m-n-2}}{f (m+1) (n+1) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^n*((b*c*f + a*d*f + a*d*f*m + b*c*f*n)/(b*d*(2 + m + n)) + f*x)^(-3 - m - n),x]

[Out]

(b*d*(2 + m + n)*(a + b*x)^(1 + m)*(c + d*x)^(1 + n)*((f*(a*d*(1 + m) + b*c*(1 +
n)))/(b*d*(2 + m + n)) + f*x)^(-2 - m - n))/((b*c - a*d)^2*f*(1 + m)*(1 + n))

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Rubi in Sympy [A]  time = 22.9425, size = 78, normalized size = 0.89 \[ \frac{b d \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (f x + \frac{f \left (a d m + a d + b c n + b c\right )}{b d \left (m + n + 2\right )}\right )^{- m - n - 2} \left (m + n + 2\right )}{f \left (m + 1\right ) \left (n + 1\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**n*((a*d*f*m+b*c*f*n+a*d*f+b*c*f)/b/d/(2+m+n)+f*x)**(-3-m-n),x)

[Out]

b*d*(a + b*x)**(m + 1)*(c + d*x)**(n + 1)*(f*x + f*(a*d*m + a*d + b*c*n + b*c)/(
b*d*(m + n + 2)))**(-m - n - 2)*(m + n + 2)/(f*(m + 1)*(n + 1)*(a*d - b*c)**2)

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Mathematica [C]  time = 74.2053, size = 5681, normalized size = 64.56 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^n*((b*c*f + a*d*f + a*d*f*m + b*c*f*n)/(b*d*(2 + m + n)) + f*x)^(-3 - m - n),x]

[Out]

Result too large to show

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Maple [B]  time = 0.01, size = 198, normalized size = 2.3 \[{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{1+n} \left ( bdxm+bdxn+adm+bcn+2\,bdx+ad+bc \right ) }{{a}^{2}{d}^{2}mn-2\,abcdmn+{b}^{2}{c}^{2}mn+{a}^{2}{d}^{2}m+{a}^{2}{d}^{2}n-2\,abcdm-2\,abcdn+{b}^{2}{c}^{2}m+{b}^{2}{c}^{2}n+{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}} \left ({\frac{f \left ( bdxm+bdxn+adm+bcn+2\,bdx+ad+bc \right ) }{bd \left ( 2+m+n \right ) }} \right ) ^{-3-m-n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*((a*d*f*m+b*c*f*n+a*d*f+b*c*f)/b/d/(2+m+n)+f*x)^(-3-m-n),x)

[Out]

(b*x+a)^(1+m)*(d*x+c)^(1+n)*(b*d*m*x+b*d*n*x+a*d*m+b*c*n+2*b*d*x+a*d+b*c)/(a^2*d
^2*m*n-2*a*b*c*d*m*n+b^2*c^2*m*n+a^2*d^2*m+a^2*d^2*n-2*a*b*c*d*m-2*a*b*c*d*n+b^2
*c^2*m+b^2*c^2*n+a^2*d^2-2*a*b*c*d+b^2*c^2)*(f*(b*d*m*x+b*d*n*x+a*d*m+b*c*n+2*b*
d*x+a*d+b*c)/b/d/(2+m+n))^(-3-m-n)

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Maxima [A]  time = 4.00643, size = 1380, normalized size = 15.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + (a*d*f*m + b*c*f*n + b*c*f + a*d*f)/(b*d*(m + n + 2)))^(-m - n - 3),x, algorithm="maxima")

[Out]

((m^3 + 3*m^2*(n + 2) + n^3 + 3*(n^2 + 4*n + 4)*m + 6*n^2 + 12*n + 8)*a*b^(m + n
+ 3)*c*d^(m + n + 3)*(m + n + 2)^(m + n) + (m^3 + 3*m^2*(n + 2) + n^3 + 3*(n^2
+ 4*n + 4)*m + 6*n^2 + 12*n + 8)*b^(m + n + 4)*d^(m + n + 4)*(m + n + 2)^(m + n)
*x^2 + ((m^3 + 3*m^2*(n + 2) + n^3 + 3*(n^2 + 4*n + 4)*m + 6*n^2 + 12*n + 8)*a*b
^(m + n + 3)*d^(m + n + 4) + (m^3 + 3*m^2*(n + 2) + n^3 + 3*(n^2 + 4*n + 4)*m +
6*n^2 + 12*n + 8)*b^(m + n + 4)*c*d^(m + n + 3))*(m + n + 2)^(m + n)*x)*e^(-m*lo
g(a*d*m + b*c*n + b*c + a*d + (b*d*m + b*d*n + 2*b*d)*x) - n*log(a*d*m + b*c*n +
b*c + a*d + (b*d*m + b*d*n + 2*b*d)*x) + m*log(b*x + a) + n*log(d*x + c))/((n^3
+ (n^3 + 3*n^2 + 3*n + 1)*m + 3*n^2 + 3*n + 1)*b^4*c^4*f^(m + n + 3) + 2*((n^2
+ 2*n + 1)*m^2 - n^3 - (n^3 + n^2 - n - 1)*m - 2*n^2 - n)*a*b^3*c^3*d*f^(m + n +
3) + (m^3*(n + 1) - (4*n^2 + 5*n + 1)*m^2 + n^3 + (n^3 - 5*n^2 - 10*n - 4)*m -
n^2 - 4*n - 2)*a^2*b^2*c^2*d^2*f^(m + n + 3) - 2*(m^3*(n + 1) - (n^2 - n - 2)*m^
2 - (2*n^2 + n - 1)*m - n^2 - n)*a^3*b*c*d^3*f^(m + n + 3) + (m^3*(n + 1) + 3*m^
2*(n + 1) + 3*m*(n + 1) + n + 1)*a^4*d^4*f^(m + n + 3) + ((m^3*(n + 1) + (2*n^2
+ 7*n + 5)*m^2 + n^3 + (n^3 + 7*n^2 + 14*n + 8)*m + 5*n^2 + 8*n + 4)*b^4*c^2*d^2
*f^(m + n + 3) - 2*(m^3*(n + 1) + (2*n^2 + 7*n + 5)*m^2 + n^3 + (n^3 + 7*n^2 + 1
4*n + 8)*m + 5*n^2 + 8*n + 4)*a*b^3*c*d^3*f^(m + n + 3) + (m^3*(n + 1) + (2*n^2
+ 7*n + 5)*m^2 + n^3 + (n^3 + 7*n^2 + 14*n + 8)*m + 5*n^2 + 8*n + 4)*a^2*b^2*d^4
*f^(m + n + 3))*x^2 + 2*(((n^2 + 2*n + 1)*m^2 + n^3 + (n^3 + 5*n^2 + 7*n + 3)*m
+ 4*n^2 + 5*n + 2)*b^4*c^3*d*f^(m + n + 3) + (m^3*(n + 1) - (n^2 - n - 2)*m^2 -
2*n^3 - (2*n^3 + 8*n^2 + 7*n + 1)*m - 7*n^2 - 7*n - 2)*a*b^3*c^2*d^2*f^(m + n +
3) - (2*m^3*(n + 1) + (n^2 + 8*n + 7)*m^2 - n^3 - (n^3 + n^2 - 7*n - 7)*m - 2*n^
2 + n + 2)*a^2*b^2*c*d^3*f^(m + n + 3) + (m^3*(n + 1) + (n^2 + 5*n + 4)*m^2 + (2
*n^2 + 7*n + 5)*m + n^2 + 3*n + 2)*a^3*b*d^4*f^(m + n + 3))*x)

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Fricas [A]  time = 0.33788, size = 448, normalized size = 5.09 \[ \frac{{\left (a^{2} c d m + a b c^{2} n + a b c^{2} + a^{2} c d +{\left (b^{2} d^{2} m + b^{2} d^{2} n + 2 \, b^{2} d^{2}\right )} x^{3} +{\left (3 \, b^{2} c d + 3 \, a b d^{2} +{\left (b^{2} c d + 2 \, a b d^{2}\right )} m +{\left (2 \, b^{2} c d + a b d^{2}\right )} n\right )} x^{2} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} +{\left (2 \, a b c d + a^{2} d^{2}\right )} m +{\left (b^{2} c^{2} + 2 \, a b c d\right )} n\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n} \left (\frac{a d f m + b c f n +{\left (b c + a d\right )} f +{\left (b d f m + b d f n + 2 \, b d f\right )} x}{b d m + b d n + 2 \, b d}\right )^{-m - n - 3}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + (a*d*f*m + b*c*f*n + b*c*f + a*d*f)/(b*d*(m + n + 2)))^(-m - n - 3),x, algorithm="fricas")

[Out]

(a^2*c*d*m + a*b*c^2*n + a*b*c^2 + a^2*c*d + (b^2*d^2*m + b^2*d^2*n + 2*b^2*d^2)
*x^3 + (3*b^2*c*d + 3*a*b*d^2 + (b^2*c*d + 2*a*b*d^2)*m + (2*b^2*c*d + a*b*d^2)*
n)*x^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2 + (2*a*b*c*d + a^2*d^2)*m + (b^2*c^2 + 2
*a*b*c*d)*n)*x)*(b*x + a)^m*(d*x + c)^n*((a*d*f*m + b*c*f*n + (b*c + a*d)*f + (b
*d*f*m + b*d*f*n + 2*b*d*f)*x)/(b*d*m + b*d*n + 2*b*d))^(-m - n - 3)/(b^2*c^2 -
2*a*b*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*m + (b^2*c^2 - 2*a*b*c*d +
a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*m)*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m*(d*x+c)**n*((a*d*f*m+b*c*f*n+a*d*f+b*c*f)/b/d/(2+m+n)+f*x)**(-3-m-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + \frac{a d f m + b c f n + b c f + a d f}{b d{\left (m + n + 2\right )}}\right )}^{-m - n - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + (a*d*f*m + b*c*f*n + b*c*f + a*d*f)/(b*d*(m + n + 2)))^(-m - n - 3),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + (a*d*f*m + b*c*f*n + b*c*f + a*d*f)/(b*
d*(m + n + 2)))^(-m - n - 3), x)