3.156 \(\int x (a+b x)^n \left (c+d x^3\right )^2 \, dx\)

Optimal. Leaf size=248 \[ -\frac{a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac{d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac{3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac{7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^2 (a+b x)^{n+8}}{b^8 (n+8)} \]

[Out]

-((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3*d)*(b
^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a
+ b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(a + b*x)^(4 + n))/(b
^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (21*a^2
*d^2*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*(a + b*x)^(7 + n))/(b^8*(7 + n)
) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + n))

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Rubi [A]  time = 0.317461, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac{d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac{3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac{7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^2 (a+b x)^{n+8}}{b^8 (n+8)} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

-((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3*d)*(b
^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a
+ b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(a + b*x)^(4 + n))/(b
^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (21*a^2
*d^2*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*(a + b*x)^(7 + n))/(b^8*(7 + n)
) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + n))

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Rubi in Sympy [A]  time = 68.5241, size = 228, normalized size = 0.92 \[ \frac{21 a^{2} d^{2} \left (a + b x\right )^{n + 6}}{b^{8} \left (n + 6\right )} - \frac{3 a^{2} d \left (a + b x\right )^{n + 3} \left (7 a^{3} d - 4 b^{3} c\right )}{b^{8} \left (n + 3\right )} - \frac{7 a d^{2} \left (a + b x\right )^{n + 7}}{b^{8} \left (n + 7\right )} + \frac{a d \left (a + b x\right )^{n + 4} \left (35 a^{3} d - 8 b^{3} c\right )}{b^{8} \left (n + 4\right )} - \frac{a \left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )^{2}}{b^{8} \left (n + 1\right )} + \frac{d^{2} \left (a + b x\right )^{n + 8}}{b^{8} \left (n + 8\right )} - \frac{d \left (a + b x\right )^{n + 5} \left (35 a^{3} d - 2 b^{3} c\right )}{b^{8} \left (n + 5\right )} + \frac{\left (a + b x\right )^{n + 2} \left (a^{3} d - b^{3} c\right ) \left (7 a^{3} d - b^{3} c\right )}{b^{8} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**n*(d*x**3+c)**2,x)

[Out]

21*a**2*d**2*(a + b*x)**(n + 6)/(b**8*(n + 6)) - 3*a**2*d*(a + b*x)**(n + 3)*(7*
a**3*d - 4*b**3*c)/(b**8*(n + 3)) - 7*a*d**2*(a + b*x)**(n + 7)/(b**8*(n + 7)) +
a*d*(a + b*x)**(n + 4)*(35*a**3*d - 8*b**3*c)/(b**8*(n + 4)) - a*(a + b*x)**(n
+ 1)*(a**3*d - b**3*c)**2/(b**8*(n + 1)) + d**2*(a + b*x)**(n + 8)/(b**8*(n + 8)
) - d*(a + b*x)**(n + 5)*(35*a**3*d - 2*b**3*c)/(b**8*(n + 5)) + (a + b*x)**(n +
2)*(a**3*d - b**3*c)*(7*a**3*d - b**3*c)/(b**8*(n + 2))

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Mathematica [A]  time = 0.485125, size = 406, normalized size = 1.64 \[ \frac{(a+b x)^{n+1} \left (-5040 a^7 d^2+5040 a^6 b d^2 (n+1) x-2520 a^5 b^2 d^2 \left (n^2+3 n+2\right ) x^2+24 a^4 b^3 d \left (2 c \left (n^3+21 n^2+146 n+336\right )+35 d \left (n^3+6 n^2+11 n+6\right ) x^3\right )-6 a^3 b^4 d (n+1) x \left (8 c \left (n^3+21 n^2+146 n+336\right )+35 d \left (n^3+9 n^2+26 n+24\right ) x^3\right )+6 a^2 b^5 d \left (n^2+3 n+2\right ) x^2 \left (4 c \left (n^3+21 n^2+146 n+336\right )+7 d \left (n^3+12 n^2+47 n+60\right ) x^3\right )-a b^6 \left (n^2+9 n+18\right ) \left (c^2 \left (n^4+24 n^3+211 n^2+804 n+1120\right )+8 c d \left (n^4+18 n^3+103 n^2+198 n+112\right ) x^3+7 d^2 \left (n^4+12 n^3+49 n^2+78 n+40\right ) x^6\right )+b^7 \left (n^5+21 n^4+165 n^3+595 n^2+954 n+504\right ) x \left (c^2 \left (n^2+13 n+40\right )+2 c d \left (n^2+10 n+16\right ) x^3+d^2 \left (n^2+7 n+10\right ) x^6\right )\right )}{b^8 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

((a + b*x)^(1 + n)*(-5040*a^7*d^2 + 5040*a^6*b*d^2*(1 + n)*x - 2520*a^5*b^2*d^2*
(2 + 3*n + n^2)*x^2 + 24*a^4*b^3*d*(2*c*(336 + 146*n + 21*n^2 + n^3) + 35*d*(6 +
11*n + 6*n^2 + n^3)*x^3) - 6*a^3*b^4*d*(1 + n)*x*(8*c*(336 + 146*n + 21*n^2 + n
^3) + 35*d*(24 + 26*n + 9*n^2 + n^3)*x^3) + 6*a^2*b^5*d*(2 + 3*n + n^2)*x^2*(4*c
*(336 + 146*n + 21*n^2 + n^3) + 7*d*(60 + 47*n + 12*n^2 + n^3)*x^3) + b^7*(504 +
954*n + 595*n^2 + 165*n^3 + 21*n^4 + n^5)*x*(c^2*(40 + 13*n + n^2) + 2*c*d*(16
+ 10*n + n^2)*x^3 + d^2*(10 + 7*n + n^2)*x^6) - a*b^6*(18 + 9*n + n^2)*(c^2*(112
0 + 804*n + 211*n^2 + 24*n^3 + n^4) + 8*c*d*(112 + 198*n + 103*n^2 + 18*n^3 + n^
4)*x^3 + 7*d^2*(40 + 78*n + 49*n^2 + 12*n^3 + n^4)*x^6)))/(b^8*(1 + n)*(2 + n)*(
3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n))

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Maple [B]  time = 0.02, size = 1142, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n*(d*x^3+c)^2,x)

[Out]

-(b*x+a)^(1+n)*(-b^7*d^2*n^7*x^7-28*b^7*d^2*n^6*x^7+7*a*b^6*d^2*n^6*x^6-322*b^7*
d^2*n^5*x^7+147*a*b^6*d^2*n^5*x^6-2*b^7*c*d*n^7*x^4-1960*b^7*d^2*n^4*x^7-42*a^2*
b^5*d^2*n^5*x^5+1225*a*b^6*d^2*n^4*x^6-62*b^7*c*d*n^6*x^4-6769*b^7*d^2*n^3*x^7-6
30*a^2*b^5*d^2*n^4*x^5+8*a*b^6*c*d*n^6*x^3+5145*a*b^6*d^2*n^3*x^6-782*b^7*c*d*n^
5*x^4-13132*b^7*d^2*n^2*x^7+210*a^3*b^4*d^2*n^4*x^4-3570*a^2*b^5*d^2*n^3*x^5+216
*a*b^6*c*d*n^5*x^3+11368*a*b^6*d^2*n^2*x^6-b^7*c^2*n^7*x-5162*b^7*c*d*n^4*x^4-13
068*b^7*d^2*n*x^7+2100*a^3*b^4*d^2*n^3*x^4-24*a^2*b^5*c*d*n^5*x^2-9450*a^2*b^5*d
^2*n^2*x^5+2264*a*b^6*c*d*n^4*x^3+12348*a*b^6*d^2*n*x^6-34*b^7*c^2*n^6*x-19088*b
^7*c*d*n^3*x^4-5040*b^7*d^2*x^7-840*a^4*b^3*d^2*n^3*x^3+7350*a^3*b^4*d^2*n^2*x^4
-576*a^2*b^5*c*d*n^4*x^2-11508*a^2*b^5*d^2*n*x^5+a*b^6*c^2*n^6+11592*a*b^6*c*d*n
^3*x^3+5040*a*b^6*d^2*x^6-478*b^7*c^2*n^5*x-39128*b^7*c*d*n^2*x^4-5040*a^4*b^3*d
^2*n^2*x^3+48*a^3*b^4*c*d*n^4*x+10500*a^3*b^4*d^2*n*x^4-5064*a^2*b^5*c*d*n^3*x^2
-5040*a^2*b^5*d^2*x^5+33*a*b^6*c^2*n^5+29984*a*b^6*c*d*n^2*x^3-3580*b^7*c^2*n^4*
x-40608*b^7*c*d*n*x^4+2520*a^5*b^2*d^2*n^2*x^2-9240*a^4*b^3*d^2*n*x^3+1056*a^3*b
^4*c*d*n^3*x+5040*a^3*b^4*d^2*x^4-19584*a^2*b^5*c*d*n^2*x^2+445*a*b^6*c^2*n^4+36
576*a*b^6*c*d*n*x^3-15289*b^7*c^2*n^3*x-16128*b^7*c*d*x^4+7560*a^5*b^2*d^2*n*x^2
-48*a^4*b^3*c*d*n^3-5040*a^4*b^3*d^2*x^3+8016*a^3*b^4*c*d*n^2*x-31200*a^2*b^5*c*
d*n*x^2+3135*a*b^6*c^2*n^3+16128*a*b^6*c*d*x^3-36706*b^7*c^2*n^2*x-5040*a^6*b*d^
2*n*x+5040*a^5*b^2*d^2*x^2-1008*a^4*b^3*c*d*n^2+23136*a^3*b^4*c*d*n*x-16128*a^2*
b^5*c*d*x^2+12154*a*b^6*c^2*n^2-44712*b^7*c^2*n*x-5040*a^6*b*d^2*x-7008*a^4*b^3*
c*d*n+16128*a^3*b^4*c*d*x+24552*a*b^6*c^2*n-20160*b^7*c^2*x+5040*a^7*d^2-16128*a
^4*b^3*c*d+20160*a*b^6*c^2)/b^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3
+118124*n^2+109584*n+40320)

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Maxima [A]  time = 0.715469, size = 640, normalized size = 2.58 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{2 \,{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac{{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} +{\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \,{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \,{\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^2*(b*x + a)^n*x,x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^
4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^
4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x
+ 24*a^5)*(b*x + a)^n*c*d/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5)
+ ((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b
^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*
x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42*(n^
5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*(n^4 + 6*n^3 + 11*n^2 + 6
*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x
^2 + 5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^2/((n^8 + 36*n^7 + 546*n^6 + 4536*
n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^8)

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Fricas [A]  time = 0.2914, size = 1642, normalized size = 6.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^2*(b*x + a)^n*x,x, algorithm="fricas")

[Out]

-(a^2*b^6*c^2*n^6 + 33*a^2*b^6*c^2*n^5 + 445*a^2*b^6*c^2*n^4 + 20160*a^2*b^6*c^2
- 16128*a^5*b^3*c*d + 5040*a^8*d^2 - (b^8*d^2*n^7 + 28*b^8*d^2*n^6 + 322*b^8*d^
2*n^5 + 1960*b^8*d^2*n^4 + 6769*b^8*d^2*n^3 + 13132*b^8*d^2*n^2 + 13068*b^8*d^2*
n + 5040*b^8*d^2)*x^8 - (a*b^7*d^2*n^7 + 21*a*b^7*d^2*n^6 + 175*a*b^7*d^2*n^5 +
735*a*b^7*d^2*n^4 + 1624*a*b^7*d^2*n^3 + 1764*a*b^7*d^2*n^2 + 720*a*b^7*d^2*n)*x
^7 + 7*(a^2*b^6*d^2*n^6 + 15*a^2*b^6*d^2*n^5 + 85*a^2*b^6*d^2*n^4 + 225*a^2*b^6*
d^2*n^3 + 274*a^2*b^6*d^2*n^2 + 120*a^2*b^6*d^2*n)*x^6 - 2*(b^8*c*d*n^7 + 31*b^8
*c*d*n^6 + 8064*b^8*c*d + (391*b^8*c*d + 21*a^3*b^5*d^2)*n^5 + (2581*b^8*c*d + 2
10*a^3*b^5*d^2)*n^4 + (9544*b^8*c*d + 735*a^3*b^5*d^2)*n^3 + 2*(9782*b^8*c*d + 5
25*a^3*b^5*d^2)*n^2 + 72*(282*b^8*c*d + 7*a^3*b^5*d^2)*n)*x^5 - 2*(a*b^7*c*d*n^7
+ 27*a*b^7*c*d*n^6 + 283*a*b^7*c*d*n^5 + 21*(69*a*b^7*c*d - 5*a^4*b^4*d^2)*n^4
+ 2*(1874*a*b^7*c*d - 315*a^4*b^4*d^2)*n^3 + 3*(1524*a*b^7*c*d - 385*a^4*b^4*d^2
)*n^2 + 126*(16*a*b^7*c*d - 5*a^4*b^4*d^2)*n)*x^4 + 3*(1045*a^2*b^6*c^2 - 16*a^5
*b^3*c*d)*n^3 + 8*(a^2*b^6*c*d*n^6 + 24*a^2*b^6*c*d*n^5 + 211*a^2*b^6*c*d*n^4 +
3*(272*a^2*b^6*c*d - 35*a^5*b^3*d^2)*n^3 + 5*(260*a^2*b^6*c*d - 63*a^5*b^3*d^2)*
n^2 + 42*(16*a^2*b^6*c*d - 5*a^5*b^3*d^2)*n)*x^3 + 2*(6077*a^2*b^6*c^2 - 504*a^5
*b^3*c*d)*n^2 - (b^8*c^2*n^7 + 34*b^8*c^2*n^6 + 20160*b^8*c^2 + 2*(239*b^8*c^2 +
12*a^3*b^5*c*d)*n^5 + 4*(895*b^8*c^2 + 132*a^3*b^5*c*d)*n^4 + (15289*b^8*c^2 +
4008*a^3*b^5*c*d)*n^3 + 2*(18353*b^8*c^2 + 5784*a^3*b^5*c*d - 1260*a^6*b^2*d^2)*
n^2 + 72*(621*b^8*c^2 + 112*a^3*b^5*c*d - 35*a^6*b^2*d^2)*n)*x^2 + 24*(1023*a^2*
b^6*c^2 - 292*a^5*b^3*c*d)*n - (a*b^7*c^2*n^7 + 33*a*b^7*c^2*n^6 + 445*a*b^7*c^2
*n^5 + 3*(1045*a*b^7*c^2 - 16*a^4*b^4*c*d)*n^4 + 2*(6077*a*b^7*c^2 - 504*a^4*b^4
*c*d)*n^3 + 24*(1023*a*b^7*c^2 - 292*a^4*b^4*c*d)*n^2 + 1008*(20*a*b^7*c^2 - 16*
a^4*b^4*c*d + 5*a^7*b*d^2)*n)*x)*(b*x + a)^n/(b^8*n^8 + 36*b^8*n^7 + 546*b^8*n^6
+ 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 109584*b^8*n
+ 40320*b^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(x*(b*x+a)**n*(d*x**3+c)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271241, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^2*(b*x + a)^n*x,x, algorithm="giac")

[Out]

Done