3.361 \(\int (a+b x)^n \left (c+d x^2\right )^3 \, dx\)
Optimal. Leaf size=223 \[ -\frac{4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac{3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
[Out]
((b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (6*a*d*(b^2*c + a^2*d)^2*(
a + b*x)^(2 + n))/(b^7*(2 + n)) + (3*d*(b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*
x)^(3 + n))/(b^7*(3 + n)) - (4*a*d^2*(3*b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7
*(4 + n)) + (3*d^2*(b^2*c + 5*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^3
*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^3*(a + b*x)^(7 + n))/(b^7*(7 + n))
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Rubi [A] time = 0.26396, antiderivative size = 223, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059
\[ -\frac{4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac{3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (6*a*d*(b^2*c + a^2*d)^2*(
a + b*x)^(2 + n))/(b^7*(2 + n)) + (3*d*(b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*
x)^(3 + n))/(b^7*(3 + n)) - (4*a*d^2*(3*b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7
*(4 + n)) + (3*d^2*(b^2*c + 5*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^3
*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^3*(a + b*x)^(7 + n))/(b^7*(7 + n))
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Rubi in Sympy [A] time = 60.525, size = 207, normalized size = 0.93 \[ - \frac{6 a d^{3} \left (a + b x\right )^{n + 6}}{b^{7} \left (n + 6\right )} - \frac{4 a d^{2} \left (a + b x\right )^{n + 4} \left (5 a^{2} d + 3 b^{2} c\right )}{b^{7} \left (n + 4\right )} - \frac{6 a d \left (a + b x\right )^{n + 2} \left (a^{2} d + b^{2} c\right )^{2}}{b^{7} \left (n + 2\right )} + \frac{d^{3} \left (a + b x\right )^{n + 7}}{b^{7} \left (n + 7\right )} + \frac{3 d^{2} \left (a + b x\right )^{n + 5} \left (5 a^{2} d + b^{2} c\right )}{b^{7} \left (n + 5\right )} + \frac{3 d \left (a + b x\right )^{n + 3} \left (a^{2} d + b^{2} c\right ) \left (5 a^{2} d + b^{2} c\right )}{b^{7} \left (n + 3\right )} + \frac{\left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )^{3}}{b^{7} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x**2+c)**3,x)
[Out]
-6*a*d**3*(a + b*x)**(n + 6)/(b**7*(n + 6)) - 4*a*d**2*(a + b*x)**(n + 4)*(5*a**
2*d + 3*b**2*c)/(b**7*(n + 4)) - 6*a*d*(a + b*x)**(n + 2)*(a**2*d + b**2*c)**2/(
b**7*(n + 2)) + d**3*(a + b*x)**(n + 7)/(b**7*(n + 7)) + 3*d**2*(a + b*x)**(n +
5)*(5*a**2*d + b**2*c)/(b**7*(n + 5)) + 3*d*(a + b*x)**(n + 3)*(a**2*d + b**2*c)
*(5*a**2*d + b**2*c)/(b**7*(n + 3)) + (a + b*x)**(n + 1)*(a**2*d + b**2*c)**3/(b
**7*(n + 1))
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Mathematica [A] time = 0.408894, size = 413, normalized size = 1.85 \[ \frac{(a+b x)^{n+1} \left (720 a^6 d^3-720 a^5 b d^3 (n+1) x+72 a^4 b^2 d^2 \left (c \left (n^2+13 n+42\right )+5 d \left (n^2+3 n+2\right ) x^2\right )-24 a^3 b^3 d^2 (n+1) x \left (3 c \left (n^2+13 n+42\right )+5 d \left (n^2+5 n+6\right ) x^2\right )+6 a^2 b^4 d \left (c^2 \left (n^4+22 n^3+179 n^2+638 n+840\right )+6 c d \left (n^4+16 n^3+83 n^2+152 n+84\right ) x^2+5 d^2 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )-6 a b^5 d (n+1) x \left (c^2 \left (n^4+22 n^3+179 n^2+638 n+840\right )+2 c d \left (n^4+18 n^3+113 n^2+288 n+252\right ) x^2+d^2 \left (n^4+14 n^3+71 n^2+154 n+120\right ) x^4\right )+b^6 \left (n^3+12 n^2+44 n+48\right ) \left (c^3 \left (n^3+15 n^2+71 n+105\right )+3 c^2 d \left (n^3+13 n^2+47 n+35\right ) x^2+3 c d^2 \left (n^3+11 n^2+31 n+21\right ) x^4+d^3 \left (n^3+9 n^2+23 n+15\right ) x^6\right )\right )}{b^7 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*(720*a^6*d^3 - 720*a^5*b*d^3*(1 + n)*x + 72*a^4*b^2*d^2*(c*(4
2 + 13*n + n^2) + 5*d*(2 + 3*n + n^2)*x^2) - 24*a^3*b^3*d^2*(1 + n)*x*(3*c*(42 +
13*n + n^2) + 5*d*(6 + 5*n + n^2)*x^2) + 6*a^2*b^4*d*(c^2*(840 + 638*n + 179*n^
2 + 22*n^3 + n^4) + 6*c*d*(84 + 152*n + 83*n^2 + 16*n^3 + n^4)*x^2 + 5*d^2*(24 +
50*n + 35*n^2 + 10*n^3 + n^4)*x^4) - 6*a*b^5*d*(1 + n)*x*(c^2*(840 + 638*n + 17
9*n^2 + 22*n^3 + n^4) + 2*c*d*(252 + 288*n + 113*n^2 + 18*n^3 + n^4)*x^2 + d^2*(
120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4) + b^6*(48 + 44*n + 12*n^2 + n^3)*(c^3*
(105 + 71*n + 15*n^2 + n^3) + 3*c^2*d*(35 + 47*n + 13*n^2 + n^3)*x^2 + 3*c*d^2*(
21 + 31*n + 11*n^2 + n^3)*x^4 + d^3*(15 + 23*n + 9*n^2 + n^3)*x^6)))/(b^7*(1 + n
)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n))
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Maple [B] time = 0.016, size = 1140, normalized size = 5.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x^2+c)^3,x)
[Out]
(b*x+a)^(1+n)*(b^6*d^3*n^6*x^6+21*b^6*d^3*n^5*x^6-6*a*b^5*d^3*n^5*x^5+3*b^6*c*d^
2*n^6*x^4+175*b^6*d^3*n^4*x^6-90*a*b^5*d^3*n^4*x^5+69*b^6*c*d^2*n^5*x^4+735*b^6*
d^3*n^3*x^6+30*a^2*b^4*d^3*n^4*x^4-12*a*b^5*c*d^2*n^5*x^3-510*a*b^5*d^3*n^3*x^5+
3*b^6*c^2*d*n^6*x^2+621*b^6*c*d^2*n^4*x^4+1624*b^6*d^3*n^2*x^6+300*a^2*b^4*d^3*n
^3*x^4-228*a*b^5*c*d^2*n^4*x^3-1350*a*b^5*d^3*n^2*x^5+75*b^6*c^2*d*n^5*x^2+2775*
b^6*c*d^2*n^3*x^4+1764*b^6*d^3*n*x^6-120*a^3*b^3*d^3*n^3*x^3+36*a^2*b^4*c*d^2*n^
4*x^2+1050*a^2*b^4*d^3*n^2*x^4-6*a*b^5*c^2*d*n^5*x-1572*a*b^5*c*d^2*n^3*x^3-1644
*a*b^5*d^3*n*x^5+b^6*c^3*n^6+741*b^6*c^2*d*n^4*x^2+6432*b^6*c*d^2*n^2*x^4+720*b^
6*d^3*x^6-720*a^3*b^3*d^3*n^2*x^3+576*a^2*b^4*c*d^2*n^3*x^2+1500*a^2*b^4*d^3*n*x
^4-138*a*b^5*c^2*d*n^4*x-4812*a*b^5*c*d^2*n^2*x^3-720*a*b^5*d^3*x^5+27*b^6*c^3*n
^5+3657*b^6*c^2*d*n^3*x^2+7236*b^6*c*d^2*n*x^4+360*a^4*b^2*d^3*n^2*x^2-72*a^3*b^
3*c*d^2*n^3*x-1320*a^3*b^3*d^3*n*x^3+6*a^2*b^4*c^2*d*n^4+2988*a^2*b^4*c*d^2*n^2*
x^2+720*a^2*b^4*d^3*x^4-1206*a*b^5*c^2*d*n^3*x-6480*a*b^5*c*d^2*n*x^3+295*b^6*c^
3*n^4+9336*b^6*c^2*d*n^2*x^2+3024*b^6*c*d^2*x^4+1080*a^4*b^2*d^3*n*x^2-1008*a^3*
b^3*c*d^2*n^2*x-720*a^3*b^3*d^3*x^3+132*a^2*b^4*c^2*d*n^3+5472*a^2*b^4*c*d^2*n*x
^2-4902*a*b^5*c^2*d*n^2*x-3024*a*b^5*c*d^2*x^3+1665*b^6*c^3*n^3+11388*b^6*c^2*d*
n*x^2-720*a^5*b*d^3*n*x+72*a^4*b^2*c*d^2*n^2+720*a^4*b^2*d^3*x^2-3960*a^3*b^3*c*
d^2*n*x+1074*a^2*b^4*c^2*d*n^2+3024*a^2*b^4*c*d^2*x^2-8868*a*b^5*c^2*d*n*x+5104*
b^6*c^3*n^2+5040*b^6*c^2*d*x^2-720*a^5*b*d^3*x+936*a^4*b^2*c*d^2*n-3024*a^3*b^3*
c*d^2*x+3828*a^2*b^4*c^2*d*n-5040*a*b^5*c^2*d*x+8028*b^6*c^3*n+720*a^6*d^3+3024*
a^4*b^2*c*d^2+5040*a^2*b^4*c^2*d+5040*b^6*c^3)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+
6769*n^3+13132*n^2+13068*n+5040)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.286411, size = 1679, normalized size = 7.53 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n,x, algorithm="fricas")
[Out]
(a*b^6*c^3*n^6 + 27*a*b^6*c^3*n^5 + 5040*a*b^6*c^3 + 5040*a^3*b^4*c^2*d + 3024*a
^5*b^2*c*d^2 + 720*a^7*d^3 + (b^7*d^3*n^6 + 21*b^7*d^3*n^5 + 175*b^7*d^3*n^4 + 7
35*b^7*d^3*n^3 + 1624*b^7*d^3*n^2 + 1764*b^7*d^3*n + 720*b^7*d^3)*x^7 + (a*b^6*d
^3*n^6 + 15*a*b^6*d^3*n^5 + 85*a*b^6*d^3*n^4 + 225*a*b^6*d^3*n^3 + 274*a*b^6*d^3
*n^2 + 120*a*b^6*d^3*n)*x^6 + 3*(b^7*c*d^2*n^6 + 1008*b^7*c*d^2 + (23*b^7*c*d^2
- 2*a^2*b^5*d^3)*n^5 + (207*b^7*c*d^2 - 20*a^2*b^5*d^3)*n^4 + 5*(185*b^7*c*d^2 -
14*a^2*b^5*d^3)*n^3 + 4*(536*b^7*c*d^2 - 25*a^2*b^5*d^3)*n^2 + 12*(201*b^7*c*d^
2 - 4*a^2*b^5*d^3)*n)*x^5 + (295*a*b^6*c^3 + 6*a^3*b^4*c^2*d)*n^4 + 3*(a*b^6*c*d
^2*n^6 + 19*a*b^6*c*d^2*n^5 + (131*a*b^6*c*d^2 + 10*a^3*b^4*d^3)*n^4 + (401*a*b^
6*c*d^2 + 60*a^3*b^4*d^3)*n^3 + 10*(54*a*b^6*c*d^2 + 11*a^3*b^4*d^3)*n^2 + 12*(2
1*a*b^6*c*d^2 + 5*a^3*b^4*d^3)*n)*x^4 + 3*(555*a*b^6*c^3 + 44*a^3*b^4*c^2*d)*n^3
+ 3*(b^7*c^2*d*n^6 + 1680*b^7*c^2*d + (25*b^7*c^2*d - 4*a^2*b^5*c*d^2)*n^5 + (2
47*b^7*c^2*d - 64*a^2*b^5*c*d^2)*n^4 + (1219*b^7*c^2*d - 332*a^2*b^5*c*d^2 - 40*
a^4*b^3*d^3)*n^3 + 8*(389*b^7*c^2*d - 76*a^2*b^5*c*d^2 - 15*a^4*b^3*d^3)*n^2 + 4
*(949*b^7*c^2*d - 84*a^2*b^5*c*d^2 - 20*a^4*b^3*d^3)*n)*x^3 + 2*(2552*a*b^6*c^3
+ 537*a^3*b^4*c^2*d + 36*a^5*b^2*c*d^2)*n^2 + 3*(a*b^6*c^2*d*n^6 + 23*a*b^6*c^2*
d*n^5 + 3*(67*a*b^6*c^2*d + 4*a^3*b^4*c*d^2)*n^4 + (817*a*b^6*c^2*d + 168*a^3*b^
4*c*d^2)*n^3 + 2*(739*a*b^6*c^2*d + 330*a^3*b^4*c*d^2 + 60*a^5*b^2*d^3)*n^2 + 24
*(35*a*b^6*c^2*d + 21*a^3*b^4*c*d^2 + 5*a^5*b^2*d^3)*n)*x^2 + 12*(669*a*b^6*c^3
+ 319*a^3*b^4*c^2*d + 78*a^5*b^2*c*d^2)*n + (b^7*c^3*n^6 + 5040*b^7*c^3 + 3*(9*b
^7*c^3 - 2*a^2*b^5*c^2*d)*n^5 + (295*b^7*c^3 - 132*a^2*b^5*c^2*d)*n^4 + 3*(555*b
^7*c^3 - 358*a^2*b^5*c^2*d - 24*a^4*b^3*c*d^2)*n^3 + 4*(1276*b^7*c^3 - 957*a^2*b
^5*c^2*d - 234*a^4*b^3*c*d^2)*n^2 + 36*(223*b^7*c^3 - 140*a^2*b^5*c^2*d - 84*a^4
*b^3*c*d^2 - 20*a^6*b*d^3)*n)*x)*(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5
+ 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n^2 + 13068*b^7*n + 5040*b^7)
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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)**n*(d*x**2+c)**3,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.26887, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n,x, algorithm="giac")
[Out]
Done