3.1051 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((7*b*B*d + A*b*e - 8
*a*B*e)*(a + b*x)^7)/(56*e*(b*d - a*e)^2*(d + e*x)^7)

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Rubi [A]  time = 0.117163, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((7*b*B*d + A*b*e - 8
*a*B*e)*(a + b*x)^7)/(56*e*(b*d - a*e)^2*(d + e*x)^7)

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Rubi in Sympy [A]  time = 12.6572, size = 73, normalized size = 0.85 \[ - \frac{\left (a + b x\right )^{7} \left (- A b e + B \left (8 a e - 7 b d\right )\right )}{56 e \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7} \left (A e - B d\right )}{8 e \left (d + e x\right )^{8} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**9,x)

[Out]

-(a + b*x)**7*(-A*b*e + B*(8*a*e - 7*b*d))/(56*e*(d + e*x)**7*(a*e - b*d)**2) -
(a + b*x)**7*(A*e - B*d)/(8*e*(d + e*x)**8*(a*e - b*d))

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Mathematica [B]  time = 0.95043, size = 597, normalized size = 6.94 \[ -\frac{a^6 e^6 (7 A e+B (d+8 e x))+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^2 b^4 e^2 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+2 a b^5 e \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^9,x]

[Out]

-(a^6*e^6*(7*A*e + B*(d + 8*e*x)) + 2*a^5*b*e^5*(3*A*e*(d + 8*e*x) + B*(d^2 + 8*
d*e*x + 28*e^2*x^2)) + a^4*b^2*e^4*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^
3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 4*a^3*b^3*e^3*(A*e*(d^3 + 8*d^2*e*
x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*
x^3 + 70*e^4*x^4)) + a^2*b^4*e^2*(3*A*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d
*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3
+ 70*d*e^4*x^4 + 56*e^5*x^5)) + 2*a*b^5*e*(A*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2
+ 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*B*(d^6 + 8*d^5*e*x + 28*d^4*e
^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)) + b^6*(A*
e*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5
*x^5 + 28*e^6*x^6) + 7*B*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 + 70
*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)))/(56*e^8*(d + e*x)^8)

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Maple [B]  time = 0.016, size = 814, normalized size = 9.5 \[ -{\frac{b \left ( 5\,A{a}^{4}b{e}^{5}-20\,A{a}^{3}{b}^{2}d{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}-20\,Aa{b}^{4}{d}^{3}{e}^{2}+5\,A{b}^{5}{d}^{4}e+2\,B{a}^{5}{e}^{5}-15\,B{a}^{4}bd{e}^{4}+40\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}-50\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,Ba{b}^{4}{d}^{4}e-7\,B{b}^{5}{d}^{5} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{6}A{e}^{7}-6\,Ad{a}^{5}b{e}^{6}+15\,A{d}^{2}{a}^{4}{b}^{2}{e}^{5}-20\,A{d}^{3}{a}^{3}{b}^{3}{e}^{4}+15\,A{d}^{4}{a}^{2}{b}^{4}{e}^{3}-6\,A{d}^{5}a{b}^{5}{e}^{2}+A{d}^{6}{b}^{6}e-Bd{a}^{6}{e}^{6}+6\,B{d}^{2}{a}^{5}b{e}^{5}-15\,B{d}^{3}{a}^{4}{b}^{2}{e}^{4}+20\,B{d}^{4}{a}^{3}{b}^{3}{e}^{3}-15\,B{d}^{5}{a}^{2}{b}^{4}{e}^{2}+6\,B{d}^{6}a{b}^{5}e-{b}^{6}B{d}^{7}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{6\,A{a}^{5}b{e}^{6}-30\,Ad{a}^{4}{b}^{2}{e}^{5}+60\,A{d}^{2}{a}^{3}{b}^{3}{e}^{4}-60\,A{d}^{3}{a}^{2}{b}^{4}{e}^{3}+30\,A{d}^{4}a{b}^{5}{e}^{2}-6\,A{d}^{5}{b}^{6}e+{a}^{6}B{e}^{6}-12\,Bd{a}^{5}b{e}^{5}+45\,B{d}^{2}{a}^{4}{b}^{2}{e}^{4}-80\,B{d}^{3}{a}^{3}{b}^{3}{e}^{3}+75\,B{d}^{4}{a}^{2}{b}^{4}{e}^{2}-36\,B{d}^{5}a{b}^{5}e+7\,{b}^{6}B{d}^{6}}{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{4} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+5\,B{a}^{2}{e}^{2}-12\,Bdabe+7\,{b}^{2}B{d}^{2} \right ) }{{e}^{8} \left ( ex+d \right ) ^{3}}}-{\frac{B{b}^{6}}{{e}^{8} \left ( ex+d \right ) }}-{\frac{{b}^{5} \left ( Abe+6\,Bae-7\,Bbd \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}-{\frac{5\,{b}^{3} \left ( 3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{b}^{3}{d}^{2}e+4\,B{a}^{3}{e}^{3}-15\,B{a}^{2}bd{e}^{2}+18\,B{d}^{2}a{b}^{2}e-7\,{b}^{3}B{d}^{3} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2} \left ( 4\,A{a}^{3}b{e}^{4}-12\,A{a}^{2}{b}^{2}d{e}^{3}+12\,Aa{b}^{3}{d}^{2}{e}^{2}-4\,A{b}^{4}{d}^{3}e+3\,B{a}^{4}{e}^{4}-16\,B{a}^{3}bd{e}^{3}+30\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,Ba{b}^{3}{d}^{3}e+7\,B{b}^{4}{d}^{4} \right ) }{{e}^{8} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^9,x)

[Out]

-1/2*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)^6-1/8*(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)^8-1/7*(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)^7-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)^3-B*b^
6/e^8/(e*x+d)-1/2*b^5*(A*b*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^2-5/4*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)^4-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)^5

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Maxima [A]  time = 1.51874, size = 1111, normalized size = 12.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/56*(56*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 7*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)*d^6*e
+ (5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (3*B
*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6
*A*a^5*b)*d*e^6 + 28*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 56*(7*B*b^6
*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 70*(
7*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6
+ (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 56*(7*B*b^6*d^4*e^3 + (6*B*a*b^5 + A*b^
6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e
^6 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 28*(7*B*b^6*d^5*e^2 + (6*B*a*b^5 + A
*b^6)*d^4*e^3 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*
d^2*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2
+ 8*(7*B*b^6*d^6*e + (6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^
4*e^3 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
5 + (2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^16*x^8 + 8*
d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x
^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)

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Fricas [A]  time = 0.216188, size = 1111, normalized size = 12.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/56*(56*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 7*A*a^6*e^7 + (6*B*a*b^5 + A*b^6)*d^6*e
+ (5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (3*B
*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6
*A*a^5*b)*d*e^6 + 28*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 56*(7*B*b^6
*d^2*e^5 + (6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 70*(
7*B*b^6*d^3*e^4 + (6*B*a*b^5 + A*b^6)*d^2*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6
+ (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 56*(7*B*b^6*d^4*e^3 + (6*B*a*b^5 + A*b^
6)*d^3*e^4 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e
^6 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 28*(7*B*b^6*d^5*e^2 + (6*B*a*b^5 + A
*b^6)*d^4*e^3 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*
d^2*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2
+ 8*(7*B*b^6*d^6*e + (6*B*a*b^5 + A*b^6)*d^5*e^2 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^
4*e^3 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
5 + (2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^16*x^8 + 8*
d*e^15*x^7 + 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x
^3 + 28*d^6*e^10*x^2 + 8*d^7*e^9*x + d^8*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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.240644, 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)^9,x, algorithm="giac")

[Out]

Done