3.934 \(\int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx\)

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))

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Rubi [A]  time = 0.118701, 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 \[ \frac{8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac{12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac{6 a (a+b x)^{m+6}}{b (m+6)}-\frac{(a+b x)^{m+7}}{b (m+7)} \]

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))

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Rubi in Sympy [A]  time = 21.1438, size = 66, normalized size = 0.79 \[ \frac{8 a^{3} \left (a + b x\right )^{m + 4}}{b \left (m + 4\right )} - \frac{12 a^{2} \left (a + b x\right )^{m + 5}}{b \left (m + 5\right )} + \frac{6 a \left (a + b x\right )^{m + 6}}{b \left (m + 6\right )} - \frac{\left (a + b x\right )^{m + 7}}{b \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(-b**2*x**2+a**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.117295, size = 114, normalized size = 1.36 \[ \frac{(a+b x)^{m+4} \left (a^3 \left (m^3+21 m^2+152 m+384\right )-3 a^2 b \left (m^3+19 m^2+118 m+232\right ) x+3 a b^2 \left (m^3+17 m^2+92 m+160\right ) x^2-b^3 \left (m^3+15 m^2+74 m+120\right ) x^3\right )}{b (m+4) (m+5) (m+6) (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(4 + m)*(a^3*(384 + 152*m + 21*m^2 + m^3) - 3*a^2*b*(232 + 118*m + 19
*m^2 + m^3)*x + 3*a*b^2*(160 + 92*m + 17*m^2 + m^3)*x^2 - b^3*(120 + 74*m + 15*m
^2 + m^3)*x^3))/(b*(4 + m)*(5 + m)*(6 + m)*(7 + m))

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Maple [B]  time = 0.015, size = 178, normalized size = 2.1 \[{\frac{ \left ( bx+a \right ) ^{4+m} \left ( -{b}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{m}^{3}{x}^{2}-15\,{b}^{3}{m}^{2}{x}^{3}-3\,{a}^{2}b{m}^{3}x+51\,a{b}^{2}{m}^{2}{x}^{2}-74\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}-57\,{a}^{2}b{m}^{2}x+276\,a{b}^{2}m{x}^{2}-120\,{b}^{3}{x}^{3}+21\,{a}^{3}{m}^{2}-354\,{a}^{2}bmx+480\,a{b}^{2}{x}^{2}+152\,{a}^{3}m-696\,{a}^{2}bx+384\,{a}^{3} \right ) }{b \left ({m}^{4}+22\,{m}^{3}+179\,{m}^{2}+638\,m+840 \right ) }} \]

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+17
9*m^2+638*m+840)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233715, size = 428, normalized size = 5.1 \[ \frac{{\left (a^{7} m^{3} + 21 \, a^{7} m^{2} + 152 \, a^{7} m -{\left (b^{7} m^{3} + 15 \, b^{7} m^{2} + 74 \, b^{7} m + 120 \, b^{7}\right )} x^{7} + 384 \, a^{7} -{\left (a b^{6} m^{3} + 9 \, a b^{6} m^{2} + 20 \, a b^{6} m\right )} x^{6} + 3 \,{\left (a^{2} b^{5} m^{3} + 19 \, a^{2} b^{5} m^{2} + 102 \, a^{2} b^{5} m + 168 \, a^{2} b^{5}\right )} x^{5} + 3 \,{\left (a^{3} b^{4} m^{3} + 13 \, a^{3} b^{4} m^{2} + 32 \, a^{3} b^{4} m\right )} x^{4} - 3 \,{\left (a^{4} b^{3} m^{3} + 23 \, a^{4} b^{3} m^{2} + 162 \, a^{4} b^{3} m + 280 \, a^{4} b^{3}\right )} x^{3} - 3 \,{\left (a^{5} b^{2} m^{3} + 17 \, a^{5} b^{2} m^{2} + 76 \, a^{5} b^{2} m\right )} x^{2} +{\left (a^{6} b m^{3} + 27 \, a^{6} b m^{2} + 254 \, a^{6} b m + 840 \, a^{6} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 15.5063, size = 2059, normalized size = 24.51 \[ \text{result too large to display} \]

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)), (-3*a**3*log(a/b + x)/(3*a**3*b + 9*a**2*b**2
*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 8*a**3/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*
x**2 + 3*b**4*x**3) - 9*a**2*b*x*log(a/b + x)/(3*a**3*b + 9*a**2*b**2*x + 9*a*b*
*3*x**2 + 3*b**4*x**3) - 18*a**2*b*x/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 +
3*b**4*x**3) - 9*a*b**2*x**2*log(a/b + x)/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*
x**2 + 3*b**4*x**3) - 18*a*b**2*x**2/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 +
3*b**4*x**3) - 3*b**3*x**3*log(a/b + x)/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x*
*2 + 3*b**4*x**3), Eq(m, -7)), (6*a**3*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*
x**2) + 13*a**3/(a**2*b + 2*a*b**2*x + b**3*x**2) + 12*a**2*b*x*log(a/b + x)/(a*
*2*b + 2*a*b**2*x + b**3*x**2) + 21*a**2*b*x/(a**2*b + 2*a*b**2*x + b**3*x**2) +
6*a*b**2*x**2*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 3*a*b**2*x**2/(a
**2*b + 2*a*b**2*x + b**3*x**2) - b**3*x**3/(a**2*b + 2*a*b**2*x + b**3*x**2), E
q(m, -6)), (-24*a**3*log(a/b + x)/(2*a*b + 2*b**2*x) - 50*a**3/(2*a*b + 2*b**2*x
) - 24*a**2*b*x*log(a/b + x)/(2*a*b + 2*b**2*x) - 24*a**2*b*x/(2*a*b + 2*b**2*x)
+ 9*a*b**2*x**2/(2*a*b + 2*b**2*x) - b**3*x**3/(2*a*b + 2*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 + 8
40*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 + 6
38*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))

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GIAC/XCAS [A]  time = 0.2162, size = 818, normalized size = 9.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^7*m^3*x^7*e^(m*ln(b*x + a)) + a*b^6*m^3*x^6*e^(m*ln(b*x + a)) + 15*b^7*m^2*x
^7*e^(m*ln(b*x + a)) - 3*a^2*b^5*m^3*x^5*e^(m*ln(b*x + a)) + 9*a*b^6*m^2*x^6*e^(
m*ln(b*x + a)) + 74*b^7*m*x^7*e^(m*ln(b*x + a)) - 3*a^3*b^4*m^3*x^4*e^(m*ln(b*x
+ a)) - 57*a^2*b^5*m^2*x^5*e^(m*ln(b*x + a)) + 20*a*b^6*m*x^6*e^(m*ln(b*x + a))
+ 120*b^7*x^7*e^(m*ln(b*x + a)) + 3*a^4*b^3*m^3*x^3*e^(m*ln(b*x + a)) - 39*a^3*b
^4*m^2*x^4*e^(m*ln(b*x + a)) - 306*a^2*b^5*m*x^5*e^(m*ln(b*x + a)) + 3*a^5*b^2*m
^3*x^2*e^(m*ln(b*x + a)) + 69*a^4*b^3*m^2*x^3*e^(m*ln(b*x + a)) - 96*a^3*b^4*m*x
^4*e^(m*ln(b*x + a)) - 504*a^2*b^5*x^5*e^(m*ln(b*x + a)) - a^6*b*m^3*x*e^(m*ln(b
*x + a)) + 51*a^5*b^2*m^2*x^2*e^(m*ln(b*x + a)) + 486*a^4*b^3*m*x^3*e^(m*ln(b*x
+ a)) - a^7*m^3*e^(m*ln(b*x + a)) - 27*a^6*b*m^2*x*e^(m*ln(b*x + a)) + 228*a^5*b
^2*m*x^2*e^(m*ln(b*x + a)) + 840*a^4*b^3*x^3*e^(m*ln(b*x + a)) - 21*a^7*m^2*e^(m
*ln(b*x + a)) - 254*a^6*b*m*x*e^(m*ln(b*x + a)) - 152*a^7*m*e^(m*ln(b*x + a)) -
840*a^6*b*x*e^(m*ln(b*x + a)) - 384*a^7*e^(m*ln(b*x + a)))/(b*m^4 + 22*b*m^3 + 1
79*b*m^2 + 638*b*m + 840*b)