3.162 \(\int \frac{(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx\)

Optimal. Leaf size=358 \[ -\frac{5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac{a d \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{d \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+1}}{b^9 (n+1)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

(a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2)*(a + b*x)^(1 + n))/(b^9*(1 + n)) -
(a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x)^(2 + n))/(b^9*(2 + n)) +
(d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) +
(2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (5*a*d^2*(3
*b^3*c - 14*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n)) + (d^2*(3*b^3*c - 56*a^3*d)*
(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^3*(a + b*x)^(7 + n))/(b^9*(7 + n))
- (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 +
n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a
*(1 + n))

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Rubi [A]  time = 0.479962, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac{a d \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{d \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+1}}{b^9 (n+1)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2)*(a + b*x)^(1 + n))/(b^9*(1 + n)) -
(a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x)^(2 + n))/(b^9*(2 + n)) +
(d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) +
(2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (5*a*d^2*(3
*b^3*c - 14*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n)) + (d^2*(3*b^3*c - 56*a^3*d)*
(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^3*(a + b*x)^(7 + n))/(b^9*(7 + n))
- (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 +
n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a
*(1 + n))

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Rubi in Sympy [A]  time = 100.157, size = 338, normalized size = 0.94 \[ \frac{28 a^{2} d^{3} \left (a + b x\right )^{n + 7}}{b^{9} \left (n + 7\right )} - \frac{2 a^{2} d^{2} \left (a + b x\right )^{n + 4} \left (28 a^{3} d - 15 b^{3} c\right )}{b^{9} \left (n + 4\right )} + \frac{a^{2} d \left (a + b x\right )^{n + 1} \left (a^{6} d^{2} - 3 a^{3} b^{3} c d + 3 b^{6} c^{2}\right )}{b^{9} \left (n + 1\right )} - \frac{8 a d^{3} \left (a + b x\right )^{n + 8}}{b^{9} \left (n + 8\right )} + \frac{5 a d^{2} \left (a + b x\right )^{n + 5} \left (14 a^{3} d - 3 b^{3} c\right )}{b^{9} \left (n + 5\right )} - \frac{a d \left (a + b x\right )^{n + 2} \left (8 a^{6} d^{2} - 15 a^{3} b^{3} c d + 6 b^{6} c^{2}\right )}{b^{9} \left (n + 2\right )} + \frac{d^{3} \left (a + b x\right )^{n + 9}}{b^{9} \left (n + 9\right )} - \frac{d^{2} \left (a + b x\right )^{n + 6} \left (56 a^{3} d - 3 b^{3} c\right )}{b^{9} \left (n + 6\right )} + \frac{d \left (a + b x\right )^{n + 3} \left (28 a^{6} d^{2} - 30 a^{3} b^{3} c d + 3 b^{6} c^{2}\right )}{b^{9} \left (n + 3\right )} - \frac{c^{3} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28*a**2*d**3*(a + b*x)**(n + 7)/(b**9*(n + 7)) - 2*a**2*d**2*(a + b*x)**(n + 4)*
(28*a**3*d - 15*b**3*c)/(b**9*(n + 4)) + a**2*d*(a + b*x)**(n + 1)*(a**6*d**2 -
3*a**3*b**3*c*d + 3*b**6*c**2)/(b**9*(n + 1)) - 8*a*d**3*(a + b*x)**(n + 8)/(b**
9*(n + 8)) + 5*a*d**2*(a + b*x)**(n + 5)*(14*a**3*d - 3*b**3*c)/(b**9*(n + 5)) -
a*d*(a + b*x)**(n + 2)*(8*a**6*d**2 - 15*a**3*b**3*c*d + 6*b**6*c**2)/(b**9*(n
+ 2)) + d**3*(a + b*x)**(n + 9)/(b**9*(n + 9)) - d**2*(a + b*x)**(n + 6)*(56*a**
3*d - 3*b**3*c)/(b**9*(n + 6)) + d*(a + b*x)**(n + 3)*(28*a**6*d**2 - 30*a**3*b*
*3*c*d + 3*b**6*c**2)/(b**9*(n + 3)) - c**3*(a + b*x)**(n + 1)*hyper((1, n + 1),
(n + 2,), 1 + b*x/a)/(a*(n + 1))

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Mathematica [B]  time = 1.84004, size = 856, normalized size = 2.39 \[ (a+b x)^n \left (\frac{c^3 \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right ) \left (\frac{a}{b x}+1\right )^{-n}}{n}+\frac{3 c^2 d \left (\frac{b x}{a}+1\right )^{-n} \left (b^3 \left (n^2+3 n+2\right ) x^3 \left (\frac{b x}{a}+1\right )^n+a b^2 n (n+1) x^2 \left (\frac{b x}{a}+1\right )^n-2 a^2 b n x \left (\frac{b x}{a}+1\right )^n+2 a^3 \left (\left (\frac{b x}{a}+1\right )^n-1\right )\right )}{b^3 (n+1) (n+2) (n+3)}+\frac{3 c d^2 \left (\frac{b x}{a}+1\right )^{-n} \left (b^6 \left (n^5+15 n^4+85 n^3+225 n^2+274 n+120\right ) x^6 \left (\frac{b x}{a}+1\right )^n+a b^5 n \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^5 \left (\frac{b x}{a}+1\right )^n-5 a^2 b^4 n \left (n^3+6 n^2+11 n+6\right ) x^4 \left (\frac{b x}{a}+1\right )^n+20 a^3 b^3 n \left (n^2+3 n+2\right ) x^3 \left (\frac{b x}{a}+1\right )^n-60 a^4 b^2 n (n+1) x^2 \left (\frac{b x}{a}+1\right )^n+120 a^5 b n x \left (\frac{b x}{a}+1\right )^n-120 a^6 \left (\left (\frac{b x}{a}+1\right )^n-1\right )\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}+\frac{d^3 \left (\frac{b x}{a}+1\right )^{-n} \left (b^9 \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 ) x^9 \left (\frac{b x}{a}+1\right )^n+a b^8 n \left (n^7+28 n^6+322 n^5+1960 n^4+6769 n^3+13132 n^2+13068 n+5040\right ) x^8 \left (\frac{b x}{a}+1\right )^n-8 a^2 b^7 n \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^7 \left (\frac{b x}{a}+1\right )^n+56 a^3 b^6 n \left (n^5+15 n^4+85 n^3+225 n^2+274 n+120\right ) x^6 \left (\frac{b x}{a}+1\right )^n-336 a^4 b^5 n \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^5 \left (\frac{b x}{a}+1\right )^n+1680 a^5 b^4 n \left (n^3+6 n^2+11 n+6\right ) x^4 \left (\frac{b x}{a}+1\right )^n-6720 a^6 b^3 n \left (n^2+3 n+2\right ) x^3 \left (\frac{b x}{a}+1\right )^n+20160 a^7 b^2 n (n+1) x^2 \left (\frac{b x}{a}+1\right )^n-40320 a^8 b n x \left (\frac{b x}{a}+1\right )^n+40320 a^9 \left (\left (\frac{b x}{a}+1\right )^n-1\right )\right )}{b^9 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^n*((3*c^2*d*(-2*a^2*b*n*x*(1 + (b*x)/a)^n + a*b^2*n*(1 + n)*x^2*(1 + (
b*x)/a)^n + b^3*(2 + 3*n + n^2)*x^3*(1 + (b*x)/a)^n + 2*a^3*(-1 + (1 + (b*x)/a)^
n)))/(b^3*(1 + n)*(2 + n)*(3 + n)*(1 + (b*x)/a)^n) + (3*c*d^2*(120*a^5*b*n*x*(1
+ (b*x)/a)^n - 60*a^4*b^2*n*(1 + n)*x^2*(1 + (b*x)/a)^n + 20*a^3*b^3*n*(2 + 3*n
+ n^2)*x^3*(1 + (b*x)/a)^n - 5*a^2*b^4*n*(6 + 11*n + 6*n^2 + n^3)*x^4*(1 + (b*x)
/a)^n + a*b^5*n*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^5*(1 + (b*x)/a)^n + b^6*(1
20 + 274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5)*x^6*(1 + (b*x)/a)^n - 120*a^6*(-1
+ (1 + (b*x)/a)^n)))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(1 + (
b*x)/a)^n) + (d^3*(-40320*a^8*b*n*x*(1 + (b*x)/a)^n + 20160*a^7*b^2*n*(1 + n)*x^
2*(1 + (b*x)/a)^n - 6720*a^6*b^3*n*(2 + 3*n + n^2)*x^3*(1 + (b*x)/a)^n + 1680*a^
5*b^4*n*(6 + 11*n + 6*n^2 + n^3)*x^4*(1 + (b*x)/a)^n - 336*a^4*b^5*n*(24 + 50*n
+ 35*n^2 + 10*n^3 + n^4)*x^5*(1 + (b*x)/a)^n + 56*a^3*b^6*n*(120 + 274*n + 225*n
^2 + 85*n^3 + 15*n^4 + n^5)*x^6*(1 + (b*x)/a)^n - 8*a^2*b^7*n*(720 + 1764*n + 16
24*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6)*x^7*(1 + (b*x)/a)^n + a*b^8*n*(5040 +
13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7)*x^8*(1 + (b
*x)/a)^n + b^9*(40320 + 109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5
+ 546*n^6 + 36*n^7 + n^8)*x^9*(1 + (b*x)/a)^n + 40320*a^9*(-1 + (1 + (b*x)/a)^n
)))/(b^9*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n)*(9 + n)
*(1 + (b*x)/a)^n) + (c^3*Hypergeometric2F1[-n, -n, 1 - n, -(a/(b*x))])/(n*(1 + a
/(b*x))^n))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( d{x}^{3}+c \right ) ^{3}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x^3+c)^3/x,x)

[Out]

int((b*x+a)^n*(d*x^3+c)^3/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{3}{\left (b x + a\right )}^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^3 + c)^3*(b*x + a)^n/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{3} x^{9} + 3 \, c d^{2} x^{6} + 3 \, c^{2} d x^{3} + c^{3}\right )}{\left (b x + a\right )}^{n}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^3*x^9 + 3*c*d^2*x^6 + 3*c^2*d*x^3 + c^3)*(b*x + a)^n/x, 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)**n*(d*x**3+c)**3/x,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^3 + c)^3*(b*x + a)^n/x, x)