3.361 \(\int (a+b x)^n (c+d x^2)^3 \, dx\)
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]
(a^2*d+b^2*c)^3*(b*x+a)^(1+n)/b^7/(1+n)-6*a*d*(a^2*d+b^2*c)^2*(b*x+a)^(2+n)/b^7/(2+n)+3*d*(a^2*d+b^2*c)*(5*a^2
*d+b^2*c)*(b*x+a)^(3+n)/b^7/(3+n)-4*a*d^2*(5*a^2*d+3*b^2*c)*(b*x+a)^(4+n)/b^7/(4+n)+3*d^2*(5*a^2*d+b^2*c)*(b*x
+a)^(5+n)/b^7/(5+n)-6*a*d^3*(b*x+a)^(6+n)/b^7/(6+n)+d^3*(b*x+a)^(7+n)/b^7/(7+n)
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Rubi [A] time = 0.13, antiderivative size = 223, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used =
{697} \[ -\frac {4 a d^2 \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {3 d^2 \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac {\left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {6 a d \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac {3 d \left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac {6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^3 (a+b x)^{n+7}}{b^7 (n+7)} \]
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))
Rule 697
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]
Rubi steps
\begin {align*} \int (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (\frac {\left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^6}-\frac {6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6}+\frac {3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6}-\frac {4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6}+\frac {3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6}-\frac {6 a d^3 (a+b x)^{5+n}}{b^6}+\frac {d^3 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac {\left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^7 (1+n)}-\frac {6 a d \left (b^2 c+a^2 d\right )^2 (a+b x)^{2+n}}{b^7 (2+n)}+\frac {3 d \left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac {4 a d^2 \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {3 d^2 \left (b^2 c+5 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^3 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^3 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 347, normalized size = 1.56 \[ \frac {(a+b x)^{n+1} \left (\frac {6 \left ((n+6) \left (a^2 d+b^2 c\right ) \left (4 (n+4) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )-4 a d (n+1) (a+b x) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )+b^4 (n+1) (n+2) (n+3) (n+4) \left (c+d x^2\right )^2\right )-a d (n+1) (a+b x) \left (4 (n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-4 a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )+b^4 (n+2) (n+3) (n+4) (n+5) \left (c+d x^2\right )^2\right )\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}+\left (c+d x^2\right )^3\right )}{b (n+7)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*((c + d*x^2)^3 + (6*((b^2*c + a^2*d)*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^
2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a
*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) - a*d*(1 + n)*(a
+ b*x)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 +
n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*
(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))))/(b*(7 + n))
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fricas [B] time = 0.83, size = 1244, normalized size = 5.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,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 + 735*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*(20
1*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*(21*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 + (247*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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giac [B] time = 0.24, size = 2085, normalized size = 9.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^7*d^3*n^6*x^7 + (b*x + a)^n*a*b^6*d^3*n^6*x^6 + 21*(b*x + a)^n*b^7*d^3*n^5*x^7 + 3*(b*x + a)^n*
b^7*c*d^2*n^6*x^5 + 15*(b*x + a)^n*a*b^6*d^3*n^5*x^6 + 175*(b*x + a)^n*b^7*d^3*n^4*x^7 + 3*(b*x + a)^n*a*b^6*c
*d^2*n^6*x^4 + 69*(b*x + a)^n*b^7*c*d^2*n^5*x^5 - 6*(b*x + a)^n*a^2*b^5*d^3*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^3
*n^4*x^6 + 735*(b*x + a)^n*b^7*d^3*n^3*x^7 + 3*(b*x + a)^n*b^7*c^2*d*n^6*x^3 + 57*(b*x + a)^n*a*b^6*c*d^2*n^5*
x^4 + 621*(b*x + a)^n*b^7*c*d^2*n^4*x^5 - 60*(b*x + a)^n*a^2*b^5*d^3*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^3*n^3*x
^6 + 1624*(b*x + a)^n*b^7*d^3*n^2*x^7 + 3*(b*x + a)^n*a*b^6*c^2*d*n^6*x^2 + 75*(b*x + a)^n*b^7*c^2*d*n^5*x^3 -
12*(b*x + a)^n*a^2*b^5*c*d^2*n^5*x^3 + 393*(b*x + a)^n*a*b^6*c*d^2*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^3*n^4*x
^4 + 2775*(b*x + a)^n*b^7*c*d^2*n^3*x^5 - 210*(b*x + a)^n*a^2*b^5*d^3*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^3*n^2*
x^6 + 1764*(b*x + a)^n*b^7*d^3*n*x^7 + (b*x + a)^n*b^7*c^3*n^6*x + 69*(b*x + a)^n*a*b^6*c^2*d*n^5*x^2 + 741*(b
*x + a)^n*b^7*c^2*d*n^4*x^3 - 192*(b*x + a)^n*a^2*b^5*c*d^2*n^4*x^3 + 1203*(b*x + a)^n*a*b^6*c*d^2*n^3*x^4 + 1
80*(b*x + a)^n*a^3*b^4*d^3*n^3*x^4 + 6432*(b*x + a)^n*b^7*c*d^2*n^2*x^5 - 300*(b*x + a)^n*a^2*b^5*d^3*n^2*x^5
+ 120*(b*x + a)^n*a*b^6*d^3*n*x^6 + 720*(b*x + a)^n*b^7*d^3*x^7 + (b*x + a)^n*a*b^6*c^3*n^6 + 27*(b*x + a)^n*b
^7*c^3*n^5*x - 6*(b*x + a)^n*a^2*b^5*c^2*d*n^5*x + 603*(b*x + a)^n*a*b^6*c^2*d*n^4*x^2 + 36*(b*x + a)^n*a^3*b^
4*c*d^2*n^4*x^2 + 3657*(b*x + a)^n*b^7*c^2*d*n^3*x^3 - 996*(b*x + a)^n*a^2*b^5*c*d^2*n^3*x^3 - 120*(b*x + a)^n
*a^4*b^3*d^3*n^3*x^3 + 1620*(b*x + a)^n*a*b^6*c*d^2*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^3*n^2*x^4 + 7236*(b*x
+ a)^n*b^7*c*d^2*n*x^5 - 144*(b*x + a)^n*a^2*b^5*d^3*n*x^5 + 27*(b*x + a)^n*a*b^6*c^3*n^5 + 295*(b*x + a)^n*b^
7*c^3*n^4*x - 132*(b*x + a)^n*a^2*b^5*c^2*d*n^4*x + 2451*(b*x + a)^n*a*b^6*c^2*d*n^3*x^2 + 504*(b*x + a)^n*a^3
*b^4*c*d^2*n^3*x^2 + 9336*(b*x + a)^n*b^7*c^2*d*n^2*x^3 - 1824*(b*x + a)^n*a^2*b^5*c*d^2*n^2*x^3 - 360*(b*x +
a)^n*a^4*b^3*d^3*n^2*x^3 + 756*(b*x + a)^n*a*b^6*c*d^2*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^3*n*x^4 + 3024*(b*x +
a)^n*b^7*c*d^2*x^5 + 295*(b*x + a)^n*a*b^6*c^3*n^4 + 6*(b*x + a)^n*a^3*b^4*c^2*d*n^4 + 1665*(b*x + a)^n*b^7*c
^3*n^3*x - 1074*(b*x + a)^n*a^2*b^5*c^2*d*n^3*x - 72*(b*x + a)^n*a^4*b^3*c*d^2*n^3*x + 4434*(b*x + a)^n*a*b^6*
c^2*d*n^2*x^2 + 1980*(b*x + a)^n*a^3*b^4*c*d^2*n^2*x^2 + 360*(b*x + a)^n*a^5*b^2*d^3*n^2*x^2 + 11388*(b*x + a)
^n*b^7*c^2*d*n*x^3 - 1008*(b*x + a)^n*a^2*b^5*c*d^2*n*x^3 - 240*(b*x + a)^n*a^4*b^3*d^3*n*x^3 + 1665*(b*x + a)
^n*a*b^6*c^3*n^3 + 132*(b*x + a)^n*a^3*b^4*c^2*d*n^3 + 5104*(b*x + a)^n*b^7*c^3*n^2*x - 3828*(b*x + a)^n*a^2*b
^5*c^2*d*n^2*x - 936*(b*x + a)^n*a^4*b^3*c*d^2*n^2*x + 2520*(b*x + a)^n*a*b^6*c^2*d*n*x^2 + 1512*(b*x + a)^n*a
^3*b^4*c*d^2*n*x^2 + 360*(b*x + a)^n*a^5*b^2*d^3*n*x^2 + 5040*(b*x + a)^n*b^7*c^2*d*x^3 + 5104*(b*x + a)^n*a*b
^6*c^3*n^2 + 1074*(b*x + a)^n*a^3*b^4*c^2*d*n^2 + 72*(b*x + a)^n*a^5*b^2*c*d^2*n^2 + 8028*(b*x + a)^n*b^7*c^3*
n*x - 5040*(b*x + a)^n*a^2*b^5*c^2*d*n*x - 3024*(b*x + a)^n*a^4*b^3*c*d^2*n*x - 720*(b*x + a)^n*a^6*b*d^3*n*x
+ 8028*(b*x + a)^n*a*b^6*c^3*n + 3828*(b*x + a)^n*a^3*b^4*c^2*d*n + 936*(b*x + a)^n*a^5*b^2*c*d^2*n + 5040*(b*
x + a)^n*b^7*c^3*x + 5040*(b*x + a)^n*a*b^6*c^3 + 5040*(b*x + a)^n*a^3*b^4*c^2*d + 3024*(b*x + a)^n*a^5*b^2*c*
d^2 + 720*(b*x + a)^n*a^7*d^3)/(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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maple [B] time = 0.01, size = 1140, normalized size = 5.11 \[ \frac {\left (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 d^{3} x^{6} b^{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 \,d^{3} x^{5} b^{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}\right ) \left (b x +a \right )^{n +1}}{\left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right ) b^{7}} \]
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)^(n+1)*(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^3-51
0*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 [B] time = 0.51, size = 472, normalized size = 2.12 \[ \frac {{\left (b x + a\right )}^{n + 1} c^{3}}{b {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3,x, algorithm="maxima")
[Out]
(b*x + a)^(n + 1)*c^3/(b*(n + 1)) + 3*((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b
*x + a)^n*c^2*d/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 3*((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^2/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((n^6 + 21*n^5 + 175*n^4 + 735*
n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^
5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3*b^4*x^4 - 120*(n^3 + 3*
n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*d^3/((n^7 + 28*n^6 +
322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)
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mupad [B] time = 3.16, size = 1144, normalized size = 5.13 \[ \frac {{\left (a+b\,x\right )}^n\,\left (720\,a^7\,d^3+72\,a^5\,b^2\,c\,d^2\,n^2+936\,a^5\,b^2\,c\,d^2\,n+3024\,a^5\,b^2\,c\,d^2+6\,a^3\,b^4\,c^2\,d\,n^4+132\,a^3\,b^4\,c^2\,d\,n^3+1074\,a^3\,b^4\,c^2\,d\,n^2+3828\,a^3\,b^4\,c^2\,d\,n+5040\,a^3\,b^4\,c^2\,d+a\,b^6\,c^3\,n^6+27\,a\,b^6\,c^3\,n^5+295\,a\,b^6\,c^3\,n^4+1665\,a\,b^6\,c^3\,n^3+5104\,a\,b^6\,c^3\,n^2+8028\,a\,b^6\,c^3\,n+5040\,a\,b^6\,c^3\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {x\,{\left (a+b\,x\right )}^n\,\left (720\,a^6\,b\,d^3\,n+72\,a^4\,b^3\,c\,d^2\,n^3+936\,a^4\,b^3\,c\,d^2\,n^2+3024\,a^4\,b^3\,c\,d^2\,n+6\,a^2\,b^5\,c^2\,d\,n^5+132\,a^2\,b^5\,c^2\,d\,n^4+1074\,a^2\,b^5\,c^2\,d\,n^3+3828\,a^2\,b^5\,c^2\,d\,n^2+5040\,a^2\,b^5\,c^2\,d\,n-b^7\,c^3\,n^6-27\,b^7\,c^3\,n^5-295\,b^7\,c^3\,n^4-1665\,b^7\,c^3\,n^3-5104\,b^7\,c^3\,n^2-8028\,b^7\,c^3\,n-5040\,b^7\,c^3\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^3\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {3\,d^2\,x^5\,{\left (a+b\,x\right )}^n\,\left (-2\,d\,a^2\,n+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,d\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-40\,a^4\,d^2\,n-4\,a^2\,b^2\,c\,d\,n^3-52\,a^2\,b^2\,c\,d\,n^2-168\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^3\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,a\,d^2\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (10\,d\,a^2+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {3\,a\,d\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+156\,a^2\,b^2\,c\,d\,n+504\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^2)^3*(a + b*x)^n,x)
[Out]
((a + b*x)^n*(720*a^7*d^3 + 5040*a*b^6*c^3 + 5040*a^3*b^4*c^2*d + 3024*a^5*b^2*c*d^2 + 5104*a*b^6*c^3*n^2 + 16
65*a*b^6*c^3*n^3 + 295*a*b^6*c^3*n^4 + 27*a*b^6*c^3*n^5 + a*b^6*c^3*n^6 + 8028*a*b^6*c^3*n + 3828*a^3*b^4*c^2*
d*n + 936*a^5*b^2*c*d^2*n + 1074*a^3*b^4*c^2*d*n^2 + 72*a^5*b^2*c*d^2*n^2 + 132*a^3*b^4*c^2*d*n^3 + 6*a^3*b^4*
c^2*d*n^4))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (x*(a + b*x)^n
*(720*a^6*b*d^3*n - 8028*b^7*c^3*n - 5104*b^7*c^3*n^2 - 1665*b^7*c^3*n^3 - 295*b^7*c^3*n^4 - 27*b^7*c^3*n^5 -
b^7*c^3*n^6 - 5040*b^7*c^3 + 5040*a^2*b^5*c^2*d*n + 3024*a^4*b^3*c*d^2*n + 3828*a^2*b^5*c^2*d*n^2 + 936*a^4*b^
3*c*d^2*n^2 + 1074*a^2*b^5*c^2*d*n^3 + 72*a^4*b^3*c*d^2*n^3 + 132*a^2*b^5*c^2*d*n^4 + 6*a^2*b^5*c^2*d*n^5))/(b
^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (d^3*x^7*(a + b*x)^n*(1764*n
+ 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 +
28*n^6 + n^7 + 5040) + (3*d^2*x^5*(a + b*x)^n*(42*b^2*c + b^2*c*n^2 - 2*a^2*d*n + 13*b^2*c*n)*(50*n + 35*n^2 +
10*n^3 + n^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (3*d*
x^3*(a + b*x)^n*(3*n + n^2 + 2)*(840*b^4*c^2 - 40*a^4*d^2*n + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3
+ b^4*c^2*n^4 - 168*a^2*b^2*c*d*n - 52*a^2*b^2*c*d*n^2 - 4*a^2*b^2*c*d*n^3))/(b^4*(13068*n + 13132*n^2 + 6769
*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*d^3*n*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^
4 + n^5 + 120))/(b*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (3*a*d^2*n*x
^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(10*a^2*d + 42*b^2*c + b^2*c*n^2 + 13*b^2*c*n))/(b^3*(13068*n + 13132*
n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (3*a*d*n*x^2*(n + 1)*(a + b*x)^n*(120*a^4*d^2 +
840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 504*a^2*b^2*c*d + 156*a^2*b^2*c
*d*n + 12*a^2*b^2*c*d*n^2))/(b^5*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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