3.31.61 \(\int (a+b x)^m (c+d x)^{-m} (e+f x)^3 \, dx\) [3061]

Optimal. Leaf size=432 \[ \frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)^2}{4 b d}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+b^2 \left (30 d^2 e^2-12 c d e f (2+m)+c^2 f^2 \left (6+5 m+m^2\right )\right )-2 b d f (a d f (3-m)-b (6 d e-c f (3+m))) x\right )}{24 b^3 d^3}-\frac {\left (a^3 d^3 f^3 \left (6-11 m+6 m^2-m^3\right )-3 a^2 b d^2 f^2 \left (2-3 m+m^2\right ) (4 d e-c f (1+m))+3 a b^2 d f (1-m) \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )-b^3 \left (24 d^3 e^3-36 c d^2 e^2 f (1+m)+12 c^2 d e f^2 \left (2+3 m+m^2\right )-c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (1+m)} \]

[Out]

1/4*f*(b*x+a)^(1+m)*(d*x+c)^(1-m)*(f*x+e)^2/b/d+1/24*f*(b*x+a)^(1+m)*(d*x+c)^(1-m)*(a^2*d^2*f^2*(m^2-5*m+6)-2*
a*b*d*f*(6*d*e*(2-m)-c*f*(-m^2+3))+b^2*(30*d^2*e^2-12*c*d*e*f*(2+m)+c^2*f^2*(m^2+5*m+6))-2*b*d*f*(a*d*f*(3-m)-
b*(6*d*e-c*f*(3+m)))*x)/b^3/d^3-1/24*(a^3*d^3*f^3*(-m^3+6*m^2-11*m+6)-3*a^2*b*d^2*f^2*(m^2-3*m+2)*(4*d*e-c*f*(
1+m))+3*a*b^2*d*f*(1-m)*(12*d^2*e^2-8*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m+2))-b^3*(24*d^3*e^3-36*c*d^2*e^2*f*(1+m)+
12*c^2*d*e*f^2*(m^2+3*m+2)-c^3*f^3*(m^3+6*m^2+11*m+6)))*(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^4/d^3/(1+m)/((d*x+c)^m)

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Rubi [A]
time = 0.33, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {102, 152, 72, 71} \begin {gather*} \frac {f (a+b x)^{m+1} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+2 b d f x (-a d f (3-m)-b c f (m+3)+6 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-12 c d e f (m+2)+30 d^2 e^2\right )\right )}{24 b^3 d^3}-\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+6 m^2-11 m+6\right )-3 a^2 b d^2 f^2 \left (m^2-3 m+2\right ) (4 d e-c f (m+1))+3 a b^2 d f (1-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )-\left (b^3 \left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+12 c^2 d e f^2 \left (m^2+3 m+2\right )-36 c d^2 e^2 f (m+1)+24 d^3 e^3\right )\right )\right ) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (m+1)}+\frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{1-m}}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(e + f*x)^3)/(c + d*x)^m,x]

[Out]

(f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m)*(e + f*x)^2)/(4*b*d) + (f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m)*(a^2*d^2*
f^2*(6 - 5*m + m^2) - 2*a*b*d*f*(6*d*e*(2 - m) - c*f*(3 - m^2)) + b^2*(30*d^2*e^2 - 12*c*d*e*f*(2 + m) + c^2*f
^2*(6 + 5*m + m^2)) + 2*b*d*f*(6*b*d*e - a*d*f*(3 - m) - b*c*f*(3 + m))*x))/(24*b^3*d^3) - ((a^3*d^3*f^3*(6 -
11*m + 6*m^2 - m^3) - 3*a^2*b*d^2*f^2*(2 - 3*m + m^2)*(4*d*e - c*f*(1 + m)) + 3*a*b^2*d*f*(1 - m)*(12*d^2*e^2
- 8*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)) - b^3*(24*d^3*e^3 - 36*c*d^2*e^2*f*(1 + m) + 12*c^2*d*e*f^2*(2
+ 3*m + m^2) - c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometr
ic2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(24*b^4*d^3*(1 + m)*(c + d*x)^m)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 102

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^{-m} (e+f x)^3 \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)^2}{4 b d}+\frac {\int (a+b x)^m (c+d x)^{-m} (e+f x) (-a f (2 c f+d e (1-m))+b e (4 d e-c f (1+m))+f (6 b d e-a d f (3-m)-b c f (3+m)) x) \, dx}{4 b d}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)^2}{4 b d}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+b^2 \left (30 d^2 e^2-12 c d e f (2+m)+c^2 f^2 \left (6+5 m+m^2\right )\right )+2 b d f (6 b d e-a d f (3-m)-b c f (3+m)) x\right )}{24 b^3 d^3}-\frac {\left (a^3 d^3 f^3 \left (6-11 m+6 m^2-m^3\right )-3 a^2 b d^2 f^2 \left (2-3 m+m^2\right ) (4 d e-c f (1+m))+3 a b^2 d f (1-m) \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )-b^3 \left (24 d^3 e^3-36 c d^2 e^2 f (1+m)+12 c^2 d e f^2 \left (2+3 m+m^2\right )-c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )\right ) \int (a+b x)^m (c+d x)^{-m} \, dx}{24 b^3 d^3}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)^2}{4 b d}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+b^2 \left (30 d^2 e^2-12 c d e f (2+m)+c^2 f^2 \left (6+5 m+m^2\right )\right )+2 b d f (6 b d e-a d f (3-m)-b c f (3+m)) x\right )}{24 b^3 d^3}-\frac {\left (\left (a^3 d^3 f^3 \left (6-11 m+6 m^2-m^3\right )-3 a^2 b d^2 f^2 \left (2-3 m+m^2\right ) (4 d e-c f (1+m))+3 a b^2 d f (1-m) \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )-b^3 \left (24 d^3 e^3-36 c d^2 e^2 f (1+m)+12 c^2 d e f^2 \left (2+3 m+m^2\right )-c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )\right ) (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}{24 b^3 d^3}\\ &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)^2}{4 b d}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+b^2 \left (30 d^2 e^2-12 c d e f (2+m)+c^2 f^2 \left (6+5 m+m^2\right )\right )+2 b d f (6 b d e-a d f (3-m)-b c f (3+m)) x\right )}{24 b^3 d^3}-\frac {\left (a^3 d^3 f^3 \left (6-11 m+6 m^2-m^3\right )-3 a^2 b d^2 f^2 \left (2-3 m+m^2\right ) (4 d e-c f (1+m))+3 a b^2 d f (1-m) \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )-b^3 \left (24 d^3 e^3-36 c d^2 e^2 f (1+m)+12 c^2 d e f^2 \left (2+3 m+m^2\right )-c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )\right ) (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 )}{24 b^4 d^3 (1+m)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.53, size = 245, normalized size = 0.57 \begin {gather*} \frac {1}{4} (a+b x)^m (c+d x)^{-m} \left (6 e^2 f x^2 \left (1+\frac {b x}{a}\right )^{-m} \left (1+\frac {d x}{c}\right )^m F_1\left (2;-m,m;3;-\frac {b x}{a},-\frac {d x}{c}\right )+4 e f^2 x^3 \left (1+\frac {b x}{a}\right )^{-m} \left (1+\frac {d x}{c}\right )^m F_1\left (3;-m,m;4;-\frac {b x}{a},-\frac {d x}{c}\right )+f^3 x^4 \left (1+\frac {b x}{a}\right )^{-m} \left (1+\frac {d x}{c}\right )^m F_1\left (4;-m,m;5;-\frac {b x}{a},-\frac {d x}{c}\right )-\frac {4 e^3 \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} (c+d x) \, _2F_1\left (1-m,-m;2-m;\frac {b (c+d x)}{b c-a d}\right )}{d (-1+m)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(e + f*x)^3)/(c + d*x)^m,x]

[Out]

((a + b*x)^m*((6*e^2*f*x^2*(1 + (d*x)/c)^m*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)])/(1 + (b*x)/a)^m + (4
*e*f^2*x^3*(1 + (d*x)/c)^m*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)])/(1 + (b*x)/a)^m + (f^3*x^4*(1 + (d*x
)/c)^m*AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)])/(1 + (b*x)/a)^m - (4*e^3*(c + d*x)*Hypergeometric2F1[1 -
m, -m, 2 - m, (b*(c + d*x))/(b*c - a*d)])/(d*(-1 + m)*((d*(a + b*x))/(-(b*c) + a*d))^m)))/(4*(c + d*x)^m)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (f x +e \right )^{3} \left (d x +c \right )^{-m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(f*x+e)^3/((d*x+c)^m),x)

[Out]

int((b*x+a)^m*(f*x+e)^3/((d*x+c)^m),x)

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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((b*x+a)^m*(f*x+e)^3/((d*x+c)^m),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(f*x+e)^3/((d*x+c)^m),x, algorithm="fricas")

[Out]

integral((f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2 + e^3)*(b*x + a)^m/(d*x + c)^m, x)

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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((b*x+a)**m*(f*x+e)**3/((d*x+c)**m),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(f*x+e)^3/((d*x+c)^m),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^3\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^m} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^3*(a + b*x)^m)/(c + d*x)^m,x)

[Out]

int(((e + f*x)^3*(a + b*x)^m)/(c + d*x)^m, x)

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