3.3073 \(\int (a+b x)^m (c+d x)^{-1-m} (e+f x) \, dx\)

Optimal. Leaf size=152 \[ \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (a d f m+b (d e-c f (m+1))) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

[Out]

(-c*f+d*e)*(b*x+a)^(1+m)/d/(-a*d+b*c)/m/((d*x+c)^m)-(a*d*f*m+b*(d*e-c*f*(1+m)))*(b*x+a)^(1+m)*(b*(d*x+c)/(-a*d
+b*c))^m*hypergeom([m, 1+m],[2+m],-d*(b*x+a)/(-a*d+b*c))/b/d/(-a*d+b*c)/m/(1+m)/((d*x+c)^m)

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Rubi [A]  time = 0.08, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{d m (b c-a d)}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (a d f m-b c f (m+1)+b d e) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b d m (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-1 - m)*(e + f*x),x]

[Out]

((d*e - c*f)*(a + b*x)^(1 + m))/(d*(b*c - a*d)*m*(c + d*x)^m) - ((b*d*e + a*d*f*m - b*c*f*(1 + m))*(a + b*x)^(
1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*d*(b
*c - a*d)*m*(1 + m)*(c + d*x)^m)

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^{-1-m} (e+f x) \, dx &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-m}}{d (b c-a d) m}-\frac {(b d e+a d f m-b c f (1+m)) \int (a+b x)^m (c+d x)^{-m} \, dx}{d (b c-a d) m}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-m}}{d (b c-a d) m}-\frac {\left ((b d e+a d f m-b c f (1+m)) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-m} \, dx}{d (b c-a d) m}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-m}}{d (b c-a d) m}-\frac {(b d e+a d f m-b c f (1+m)) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{b d (b c-a d) m (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 114, normalized size = 0.75 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^m (a d f m-b c f (m+1)+b d e) \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )}{b (m+1)}+c f-d e\right )}{d m (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-1 - m)*(e + f*x),x]

[Out]

((a + b*x)^(1 + m)*(-(d*e) + c*f + ((b*d*e + a*d*f*m - b*c*f*(1 + m))*((b*(c + d*x))/(b*c - a*d))^m*Hypergeome
tric2F1[m, 1 + m, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(1 + m))))/(d*(-(b*c) + a*d)*m*(c + d*x)^m)

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="fricas")

[Out]

integral((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), x)

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{-m -1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-m-1)*(f*x+e),x)

[Out]

int((b*x+a)^m*(d*x+c)^(-m-1)*(f*x+e),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-m)*(f*x+e),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)*(a + b*x)^m)/(c + d*x)^(m + 1),x)

[Out]

int(((e + f*x)*(a + b*x)^m)/(c + d*x)^(m + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**(-1-m)*(f*x+e),x)

[Out]

Timed out

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