3.3044 \(\int (a+b x)^m (c+d x)^{-m} (e+f x)^3 \, dx\)

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

[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*(a*d*f*(3 - m) - b*(6*d*e - 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*Hypergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(24*b^
4*d^3*(1 + m)*(c + d*x)^m)

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Rubi [A]  time = 1.25691, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{f (a+b x)^{m+1} (c+d x)^{1-m} \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f \left (6 d e (2-m)-c f \left (3-m^2\right )\right )+2 b d f x (-a d f (3-m)-b c f (m+3)+6 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-12 c d e f (m+2)+30 d^2 e^2\right )\right )}{24 b^3 d^3}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+6 m^2-11 m+6\right )-3 a^2 b d^2 f^2 \left (m^2-3 m+2\right ) (4 d e-c f (m+1))+3 a b^2 d f (1-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )+b^3 \left (-\left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+12 c^2 d e f^2 \left (m^2+3 m+2\right )-36 c d^2 e^2 f (m+1)+24 d^3 e^3\right )\right )\right ) \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (m+1)}+\frac{f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{1-m}}{4 b d} \]

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*Hypergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(24*b^
4*d^3*(1 + m)*(c + d*x)^m)

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Rubi in Sympy [A]  time = 138.712, size = 556, normalized size = 1.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 1)*(e + f*x)**2/(4*b*d) - f*(a + b*x)**(m
+ 1)*(c + d*x)**(-m + 1)*(-a*d*f*(-m + 2)*(-6*b*d*e + f*(a*d*(-m + 3) + b*c*(m +
3))) - b*c*f*(m + 2)*(-6*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 3))) + 2*b*d*f*x*(-
6*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 3))) + 3*b*d*(-4*b*d*e**2 + e*(-6*b*d*e + f
*(a*d*(-m + 3) + b*c*(m + 3))) + f*(2*a*c*f + e*(a*d*(-m + 1) + b*c*(m + 1)))))/
(24*b**3*d**3) - (b*(-c - d*x)/(a*d - b*c))**m*(a + b*x)**(m + 1)*(c + d*x)**(-m
)*(a**2*d**2*f**2*(-m + 1)*(-m + 2)*(-6*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 3)))
- a*b*d*f*(-m + 1)*(-2*c*f*(m + 1)*(-6*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 3))) +
3*d*(-4*b*d*e**2 + e*(-6*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 3))) + f*(2*a*c*f +
e*(a*d*(-m + 1) + b*c*(m + 1))))) + b**2*(c**2*f**2*(m + 1)*(m + 2)*(-6*b*d*e +
f*(a*d*(-m + 3) + b*c*(m + 3))) - 3*c*d*f*(m + 1)*(-4*b*d*e**2 + e*(-6*b*d*e +
f*(a*d*(-m + 3) + b*c*(m + 3))) + f*(2*a*c*f + e*(a*d*(-m + 1) + b*c*(m + 1))))
+ 6*d**2*e*(-4*b*d*e**2 + f*(2*a*c*f + e*(a*d*(-m + 1) + b*c*(m + 1))))))*hyper(
(m, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(24*b**4*d**3*(m + 1))

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Mathematica [C]  time = 2.98784, size = 440, normalized size = 1.02 \[ (a+b x)^m (c+d x)^{-m} \left (\frac{9 a c e^2 f x^2 F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{4 a c e f^2 x^3 F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+m x \left (b c F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{5 a c f^3 x^4 F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{20 a c F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )+4 b c m x F_1\left (5;1-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )-4 a d m x F_1\left (5;-m,m+1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{e^3 (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*((9*a*c*e^2*f*x^2*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)])/(6
*a*c*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)] + 2*m*x*(b*c*AppellF1[3, 1 -
m, m, 4, -((b*x)/a), -((d*x)/c)] - a*d*AppellF1[3, -m, 1 + m, 4, -((b*x)/a), -((
d*x)/c)])) + (4*a*c*e*f^2*x^3*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)])/(4*
a*c*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)] + m*x*(b*c*AppellF1[4, 1 - m,
m, 5, -((b*x)/a), -((d*x)/c)] - a*d*AppellF1[4, -m, 1 + m, 5, -((b*x)/a), -((d*x
)/c)])) + (5*a*c*f^3*x^4*AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)])/(20*a*c*
AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)] + 4*b*c*m*x*AppellF1[5, 1 - m, m,
6, -((b*x)/a), -((d*x)/c)] - 4*a*d*m*x*AppellF1[5, -m, 1 + m, 6, -((b*x)/a), -((
d*x)/c)]) - (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)))/(c + d*x)^m

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( fx+e \right ) ^{3}}{ \left ( dx+c \right ) ^{m}}}\, dx \]

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., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m/(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., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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