### 3.945 $$\int (a+b x)^m (a^2-b^2 x^2)^3 \, dx$$

Optimal. Leaf size=84 $\frac{8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac{12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac{6 a (a+b x)^{m+6}}{b (m+6)}-\frac{(a+b x)^{m+7}}{b (m+7)}$

[Out]

(8*a^3*(a + b*x)^(4 + m))/(b*(4 + m)) - (12*a^2*(a + b*x)^(5 + m))/(b*(5 + m)) + (6*a*(a + b*x)^(6 + m))/(b*(6
+ m)) - (a + b*x)^(7 + m)/(b*(7 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.0418968, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {627, 43} $\frac{8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac{12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac{6 a (a+b x)^{m+6}}{b (m+6)}-\frac{(a+b x)^{m+7}}{b (m+7)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^m*(a^2 - b^2*x^2)^3,x]

[Out]

(8*a^3*(a + b*x)^(4 + m))/(b*(4 + m)) - (12*a^2*(a + b*x)^(5 + m))/(b*(5 + m)) + (6*a*(a + b*x)^(6 + m))/(b*(6
+ m)) - (a + b*x)^(7 + m)/(b*(7 + m))

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx &=\int (a-b x)^3 (a+b x)^{3+m} \, dx\\ &=\int \left (8 a^3 (a+b x)^{3+m}-12 a^2 (a+b x)^{4+m}+6 a (a+b x)^{5+m}-(a+b x)^{6+m}\right ) \, dx\\ &=\frac{8 a^3 (a+b x)^{4+m}}{b (4+m)}-\frac{12 a^2 (a+b x)^{5+m}}{b (5+m)}+\frac{6 a (a+b x)^{6+m}}{b (6+m)}-\frac{(a+b x)^{7+m}}{b (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0645084, size = 68, normalized size = 0.81 $\frac{(a+b x)^{m+4} \left (-\frac{12 a^2 (a+b x)}{m+5}+\frac{8 a^3}{m+4}+\frac{6 a (a+b x)^2}{m+6}-\frac{(a+b x)^3}{m+7}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^m*(a^2 - b^2*x^2)^3,x]

[Out]

((a + b*x)^(4 + m)*((8*a^3)/(4 + m) - (12*a^2*(a + b*x))/(5 + m) + (6*a*(a + b*x)^2)/(6 + m) - (a + b*x)^3/(7
+ m)))/b

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 178, normalized size = 2.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{4+m} \left ( -{b}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{m}^{3}{x}^{2}-15\,{b}^{3}{m}^{2}{x}^{3}-3\,{a}^{2}b{m}^{3}x+51\,a{b}^{2}{m}^{2}{x}^{2}-74\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}-57\,{a}^{2}b{m}^{2}x+276\,a{b}^{2}m{x}^{2}-120\,{b}^{3}{x}^{3}+21\,{a}^{3}{m}^{2}-354\,{a}^{2}bmx+480\,a{b}^{2}{x}^{2}+152\,{a}^{3}m-696\,{a}^{2}xb+384\,{a}^{3} \right ) }{b \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(-b^2*x^2+a^2)^3,x)

[Out]

(b*x+a)^(4+m)*(-b^3*m^3*x^3+3*a*b^2*m^3*x^2-15*b^3*m^2*x^3-3*a^2*b*m^3*x+51*a*b^2*m^2*x^2-74*b^3*m*x^3+a^3*m^3
-57*a^2*b*m^2*x+276*a*b^2*m*x^2-120*b^3*x^3+21*a^3*m^2-354*a^2*b*m*x+480*a*b^2*x^2+152*a^3*m-696*a^2*b*x+384*a
^3)/b/(m^4+22*m^3+179*m^2+638*m+840)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.99885, size = 682, normalized size = 8.12 \begin{align*} \frac{{\left (a^{7} m^{3} + 21 \, a^{7} m^{2} + 152 \, a^{7} m -{\left (b^{7} m^{3} + 15 \, b^{7} m^{2} + 74 \, b^{7} m + 120 \, b^{7}\right )} x^{7} + 384 \, a^{7} -{\left (a b^{6} m^{3} + 9 \, a b^{6} m^{2} + 20 \, a b^{6} m\right )} x^{6} + 3 \,{\left (a^{2} b^{5} m^{3} + 19 \, a^{2} b^{5} m^{2} + 102 \, a^{2} b^{5} m + 168 \, a^{2} b^{5}\right )} x^{5} + 3 \,{\left (a^{3} b^{4} m^{3} + 13 \, a^{3} b^{4} m^{2} + 32 \, a^{3} b^{4} m\right )} x^{4} - 3 \,{\left (a^{4} b^{3} m^{3} + 23 \, a^{4} b^{3} m^{2} + 162 \, a^{4} b^{3} m + 280 \, a^{4} b^{3}\right )} x^{3} - 3 \,{\left (a^{5} b^{2} m^{3} + 17 \, a^{5} b^{2} m^{2} + 76 \, a^{5} b^{2} m\right )} x^{2} +{\left (a^{6} b m^{3} + 27 \, a^{6} b m^{2} + 254 \, a^{6} b m + 840 \, a^{6} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

(a^7*m^3 + 21*a^7*m^2 + 152*a^7*m - (b^7*m^3 + 15*b^7*m^2 + 74*b^7*m + 120*b^7)*x^7 + 384*a^7 - (a*b^6*m^3 + 9
*a*b^6*m^2 + 20*a*b^6*m)*x^6 + 3*(a^2*b^5*m^3 + 19*a^2*b^5*m^2 + 102*a^2*b^5*m + 168*a^2*b^5)*x^5 + 3*(a^3*b^4
*m^3 + 13*a^3*b^4*m^2 + 32*a^3*b^4*m)*x^4 - 3*(a^4*b^3*m^3 + 23*a^4*b^3*m^2 + 162*a^4*b^3*m + 280*a^4*b^3)*x^3
- 3*(a^5*b^2*m^3 + 17*a^5*b^2*m^2 + 76*a^5*b^2*m)*x^2 + (a^6*b*m^3 + 27*a^6*b*m^2 + 254*a^6*b*m + 840*a^6*b)*
x)*(b*x + a)^m/(b*m^4 + 22*b*m^3 + 179*b*m^2 + 638*b*m + 840*b)

________________________________________________________________________________________

Sympy [A]  time = 5.32617, size = 2096, normalized size = 24.95 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(-b**2*x**2+a**2)**3,x)

[Out]

Piecewise((a**6*a**m*x, Eq(b, 0)), (-15*a**3*log(a/b + x)/(15*a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b*
*4*x**3) - 8*a**3/(15*a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b**4*x**3) - 45*a**2*b*x*log(a/b + x)/(15*
a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b**4*x**3) + 6*a**2*b*x/(15*a**3*b + 45*a**2*b**2*x + 45*a*b**3*
x**2 + 15*b**4*x**3) - 45*a*b**2*x**2*log(a/b + x)/(15*a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b**4*x**3
) + 6*a*b**2*x**2/(15*a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b**4*x**3) - 15*b**3*x**3*log(a/b + x)/(15
*a**3*b + 45*a**2*b**2*x + 45*a*b**3*x**2 + 15*b**4*x**3) + 32*b**3*x**3/(15*a**3*b + 45*a**2*b**2*x + 45*a*b*
*3*x**2 + 15*b**4*x**3), Eq(m, -7)), (12*a**3*log(a/b + x)/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2) + 5*a**3/(2*a
**2*b + 4*a*b**2*x + 2*b**3*x**2) + 24*a**2*b*x*log(a/b + x)/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2) + 12*a*b**2
*x**2*log(a/b + x)/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2) - 15*a*b**2*x**2/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2
) - 2*b**3*x**3/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2), Eq(m, -6)), (-48*a**3*log(a/b + x)/(4*a*b + 4*b**2*x) -
29*a**3/(4*a*b + 4*b**2*x) - 48*a**2*b*x*log(a/b + x)/(4*a*b + 4*b**2*x) + 23*a**2*b*x/(4*a*b + 4*b**2*x) + 1
8*a*b**2*x**2/(4*a*b + 4*b**2*x) - 2*b**3*x**3/(4*a*b + 4*b**2*x), Eq(m, -5)), (8*a**3*log(a/b + x)/b - 7*a**2
*x + 2*a*b*x**2 - b**2*x**3/3, Eq(m, -4)), (a**7*m**3*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m
+ 840*b) + 21*a**7*m**2*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 152*a**7*m*(a + b*x
)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 384*a**7*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m
**2 + 638*b*m + 840*b) + a**6*b*m**3*x*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 27*a
**6*b*m**2*x*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 254*a**6*b*m*x*(a + b*x)**m/(b
*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 840*a**6*b*x*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2
+ 638*b*m + 840*b) - 3*a**5*b**2*m**3*x**2*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) -
51*a**5*b**2*m**2*x**2*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - 228*a**5*b**2*m*x**
2*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - 3*a**4*b**3*m**3*x**3*(a + b*x)**m/(b*m**
4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - 69*a**4*b**3*m**2*x**3*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*
b*m**2 + 638*b*m + 840*b) - 486*a**4*b**3*m*x**3*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840
*b) - 840*a**4*b**3*x**3*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 3*a**3*b**4*m**3*x
**4*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 39*a**3*b**4*m**2*x**4*(a + b*x)**m/(b*
m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 96*a**3*b**4*m*x**4*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*
b*m**2 + 638*b*m + 840*b) + 3*a**2*b**5*m**3*x**5*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 84
0*b) + 57*a**2*b**5*m**2*x**5*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 306*a**2*b**5
*m*x**5*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 504*a**2*b**5*x**5*(a + b*x)**m/(b*
m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - a*b**6*m**3*x**6*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m
**2 + 638*b*m + 840*b) - 9*a*b**6*m**2*x**6*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) -
20*a*b**6*m*x**6*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - b**7*m**3*x**7*(a + b*x)*
*m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) - 15*b**7*m**2*x**7*(a + b*x)**m/(b*m**4 + 22*b*m**3 +
179*b*m**2 + 638*b*m + 840*b) - 74*b**7*m*x**7*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b
) - 120*b**7*x**7*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b), True))

________________________________________________________________________________________

Giac [B]  time = 1.15659, size = 740, normalized size = 8.81 \begin{align*} -\frac{{\left (b x + a\right )}^{m} b^{7} m^{3} x^{7} +{\left (b x + a\right )}^{m} a b^{6} m^{3} x^{6} + 15 \,{\left (b x + a\right )}^{m} b^{7} m^{2} x^{7} - 3 \,{\left (b x + a\right )}^{m} a^{2} b^{5} m^{3} x^{5} + 9 \,{\left (b x + a\right )}^{m} a b^{6} m^{2} x^{6} + 74 \,{\left (b x + a\right )}^{m} b^{7} m x^{7} - 3 \,{\left (b x + a\right )}^{m} a^{3} b^{4} m^{3} x^{4} - 57 \,{\left (b x + a\right )}^{m} a^{2} b^{5} m^{2} x^{5} + 20 \,{\left (b x + a\right )}^{m} a b^{6} m x^{6} + 120 \,{\left (b x + a\right )}^{m} b^{7} x^{7} + 3 \,{\left (b x + a\right )}^{m} a^{4} b^{3} m^{3} x^{3} - 39 \,{\left (b x + a\right )}^{m} a^{3} b^{4} m^{2} x^{4} - 306 \,{\left (b x + a\right )}^{m} a^{2} b^{5} m x^{5} + 3 \,{\left (b x + a\right )}^{m} a^{5} b^{2} m^{3} x^{2} + 69 \,{\left (b x + a\right )}^{m} a^{4} b^{3} m^{2} x^{3} - 96 \,{\left (b x + a\right )}^{m} a^{3} b^{4} m x^{4} - 504 \,{\left (b x + a\right )}^{m} a^{2} b^{5} x^{5} -{\left (b x + a\right )}^{m} a^{6} b m^{3} x + 51 \,{\left (b x + a\right )}^{m} a^{5} b^{2} m^{2} x^{2} + 486 \,{\left (b x + a\right )}^{m} a^{4} b^{3} m x^{3} -{\left (b x + a\right )}^{m} a^{7} m^{3} - 27 \,{\left (b x + a\right )}^{m} a^{6} b m^{2} x + 228 \,{\left (b x + a\right )}^{m} a^{5} b^{2} m x^{2} + 840 \,{\left (b x + a\right )}^{m} a^{4} b^{3} x^{3} - 21 \,{\left (b x + a\right )}^{m} a^{7} m^{2} - 254 \,{\left (b x + a\right )}^{m} a^{6} b m x - 152 \,{\left (b x + a\right )}^{m} a^{7} m - 840 \,{\left (b x + a\right )}^{m} a^{6} b x - 384 \,{\left (b x + a\right )}^{m} a^{7}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(-b^2*x^2+a^2)^3,x, algorithm="giac")

[Out]

-((b*x + a)^m*b^7*m^3*x^7 + (b*x + a)^m*a*b^6*m^3*x^6 + 15*(b*x + a)^m*b^7*m^2*x^7 - 3*(b*x + a)^m*a^2*b^5*m^3
*x^5 + 9*(b*x + a)^m*a*b^6*m^2*x^6 + 74*(b*x + a)^m*b^7*m*x^7 - 3*(b*x + a)^m*a^3*b^4*m^3*x^4 - 57*(b*x + a)^m
*a^2*b^5*m^2*x^5 + 20*(b*x + a)^m*a*b^6*m*x^6 + 120*(b*x + a)^m*b^7*x^7 + 3*(b*x + a)^m*a^4*b^3*m^3*x^3 - 39*(
b*x + a)^m*a^3*b^4*m^2*x^4 - 306*(b*x + a)^m*a^2*b^5*m*x^5 + 3*(b*x + a)^m*a^5*b^2*m^3*x^2 + 69*(b*x + a)^m*a^
4*b^3*m^2*x^3 - 96*(b*x + a)^m*a^3*b^4*m*x^4 - 504*(b*x + a)^m*a^2*b^5*x^5 - (b*x + a)^m*a^6*b*m^3*x + 51*(b*x
+ a)^m*a^5*b^2*m^2*x^2 + 486*(b*x + a)^m*a^4*b^3*m*x^3 - (b*x + a)^m*a^7*m^3 - 27*(b*x + a)^m*a^6*b*m^2*x + 2
28*(b*x + a)^m*a^5*b^2*m*x^2 + 840*(b*x + a)^m*a^4*b^3*x^3 - 21*(b*x + a)^m*a^7*m^2 - 254*(b*x + a)^m*a^6*b*m*
x - 152*(b*x + a)^m*a^7*m - 840*(b*x + a)^m*a^6*b*x - 384*(b*x + a)^m*a^7)/(b*m^4 + 22*b*m^3 + 179*b*m^2 + 638
*b*m + 840*b)