3.1052 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=135 \[ \frac{b (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac{(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(9*e*(b*d - a*e)*(d + e*x)^9) + ((7*b*B*d + 2*A*b*e -
9*a*B*e)*(a + b*x)^7)/(72*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(7*b*B*d + 2*A*b*e
- 9*a*B*e)*(a + b*x)^7)/(504*e*(b*d - a*e)^3*(d + e*x)^7)

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Rubi [A]  time = 0.1735, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac{(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(9*e*(b*d - a*e)*(d + e*x)^9) + ((7*b*B*d + 2*A*b*e -
9*a*B*e)*(a + b*x)^7)/(72*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(7*b*B*d + 2*A*b*e
- 9*a*B*e)*(a + b*x)^7)/(504*e*(b*d - a*e)^3*(d + e*x)^7)

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Rubi in Sympy [A]  time = 25.472, size = 122, normalized size = 0.9 \[ - \frac{b \left (a + b x\right )^{7} \left (2 A b e - 9 B a e + 7 B b d\right )}{504 e \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{7} \left (2 A b e - 9 B a e + 7 B b d\right )}{72 e \left (d + e x\right )^{8} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7} \left (A e - B d\right )}{9 e \left (d + e x\right )^{9} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**10,x)

[Out]

-b*(a + b*x)**7*(2*A*b*e - 9*B*a*e + 7*B*b*d)/(504*e*(d + e*x)**7*(a*e - b*d)**3
) + (a + b*x)**7*(2*A*b*e - 9*B*a*e + 7*B*b*d)/(72*e*(d + e*x)**8*(a*e - b*d)**2
) - (a + b*x)**7*(A*e - B*d)/(9*e*(d + e*x)**9*(a*e - b*d))

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Mathematica [B]  time = 0.999067, size = 603, normalized size = 4.47 \[ -\frac{7 a^6 e^6 (8 A e+B (d+9 e x))+6 a^5 b e^5 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 a^2 b^4 e^2 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+6 a b^5 e \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+b^6 \left (2 A e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 B \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{504 e^8 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-(7*a^6*e^6*(8*A*e + B*(d + 9*e*x)) + 6*a^5*b*e^5*(7*A*e*(d + 9*e*x) + 2*B*(d^2
+ 9*d*e*x + 36*e^2*x^2)) + 15*a^4*b^2*e^4*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) +
B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 4*a^3*b^3*e^3*(5*A*e*(d^3 + 9
*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 +
84*d*e^3*x^3 + 126*e^4*x^4)) + 3*a^2*b^4*e^2*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^
2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84
*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 6*a*b^5*e*(A*e*(d^5 + 9*d^4*e*x +
36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d
^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*
e^6*x^6)) + b^6*(2*A*e*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*
d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^
2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x
^7)))/(504*e^8*(d + e*x)^9)

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Maple [B]  time = 0.013, size = 814, normalized size = 6. \[ -{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{9\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{5\,{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{6\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{6\,A{a}^{5}b{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+{a}^{6}B{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{3\,b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{3\,{e}^{8} \left ( ex+d \right ) ^{3}}}-{\frac{B{b}^{6}}{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{4} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+5\,B{a}^{2}{e}^{2}-12\,Bdabe+7\,{b}^{2}B{d}^{2} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,B{d}^{2}a{b}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{{e}^{8} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^10,x)

[Out]

-1/9*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a
^2*b^4*d^4*e^3-6*A*a*b^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*
a^4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*B*a^2*b^4*d^5*e^2+6*B*a*b^5*d^6*e-B*b^6*
d^7)/e^8/(e*x+d)^9-5/6*b^2*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-
4*A*b^4*d^3*e+3*B*a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a*b^3*d^3*e
+7*B*b^4*d^4)/e^8/(e*x+d)^6-1/8*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d
^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*
b*d*e^5+45*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^
5*d^5*e+7*B*b^6*d^6)/e^8/(e*x+d)^8-3/7*b*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*
a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40
*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/e^8/(e*x+d
)^7-1/3*b^5*(A*b*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^3-1/2*B*b^6/e^8/(e*x+d)^2-3/4*b^
4*(2*A*a*b*e^2-2*A*b^2*d*e+5*B*a^2*e^2-12*B*a*b*d*e+7*B*b^2*d^2)/e^8/(e*x+d)^4-b
^3*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+4*B*a^3*e^3-15*B*a^2*b*d*e^2+18*
B*a*b^2*d^2*e-7*B*b^3*d^3)/e^8/(e*x+d)^5

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Maxima [A]  time = 1.42421, size = 1162, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^10,x, algorithm="maxima")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(6*B*a*b^5 + A*b^6)*d
^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e
^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5
+ 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*
x^6 + 126*(7*B*b^6*d^2*e^5 + 2*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*
a*b^5)*e^7)*x^5 + 126*(7*B*b^6*d^3*e^4 + 2*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^6 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 84*(7*B*b^
6*d^4*e^3 + 2*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5
+ 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 +
36*(7*B*b^6*d^5*e^2 + 2*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^
5)*d^3*e^4 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^
3)*d*e^6 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(6*B*a*b^
5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 4*(4*B*a^3*b^3 + 3*A*
a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6*(2*B*a^5*b + 5*A*a^
4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e
^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^
3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8)

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Fricas [A]  time = 0.210887, size = 1162, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(6*B*a*b^5 + A*b^6)*d
^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e
^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5
+ 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*
x^6 + 126*(7*B*b^6*d^2*e^5 + 2*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*
a*b^5)*e^7)*x^5 + 126*(7*B*b^6*d^3*e^4 + 2*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*
a^2*b^4 + 2*A*a*b^5)*d*e^6 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 84*(7*B*b^
6*d^4*e^3 + 2*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5
+ 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 +
36*(7*B*b^6*d^5*e^2 + 2*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^
5)*d^3*e^4 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^
3)*d*e^6 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(6*B*a*b^
5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 4*(4*B*a^3*b^3 + 3*A*
a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6*(2*B*a^5*b + 5*A*a^
4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e
^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^
3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8)

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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)**6*(B*x+A)/(e*x+d)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23846, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^10,x, algorithm="giac")

[Out]

Done