3.400 \(\int x^2 (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a^2 (a+b x)^{7/2} (A b-a B)}{7 b^4}+\frac{2 (a+b x)^{11/2} (A b-3 a B)}{11 b^4}-\frac{2 a (a+b x)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac{2 B (a+b x)^{13/2}}{13 b^4} \]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x)^(9/
2))/(9*b^4) + (2*(A*b - 3*a*B)*(a + b*x)^(11/2))/(11*b^4) + (2*B*(a + b*x)^(13/2
))/(13*b^4)

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Rubi [A]  time = 0.12443, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2 a^2 (a+b x)^{7/2} (A b-a B)}{7 b^4}+\frac{2 (a+b x)^{11/2} (A b-3 a B)}{11 b^4}-\frac{2 a (a+b x)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac{2 B (a+b x)^{13/2}}{13 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x)^(9/
2))/(9*b^4) + (2*(A*b - 3*a*B)*(a + b*x)^(11/2))/(11*b^4) + (2*B*(a + b*x)^(13/2
))/(13*b^4)

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Rubi in Sympy [A]  time = 18.3557, size = 92, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{13}{2}}}{13 b^{4}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{9}{2}} \left (2 A b - 3 B a\right )}{9 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{11}{2}} \left (A b - 3 B a\right )}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*B*(a + b*x)**(13/2)/(13*b**4) + 2*a**2*(a + b*x)**(7/2)*(A*b - B*a)/(7*b**4) -
2*a*(a + b*x)**(9/2)*(2*A*b - 3*B*a)/(9*b**4) + 2*(a + b*x)**(11/2)*(A*b - 3*B*
a)/(11*b**4)

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Mathematica [A]  time = 0.0992616, size = 68, normalized size = 0.72 \[ \frac{2 (a+b x)^{7/2} \left (-48 a^3 B+8 a^2 b (13 A+21 B x)-14 a b^2 x (26 A+27 B x)+63 b^3 x^2 (13 A+11 B x)\right )}{9009 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(7/2)*(-48*a^3*B + 63*b^3*x^2*(13*A + 11*B*x) + 8*a^2*b*(13*A + 21*
B*x) - 14*a*b^2*x*(26*A + 27*B*x)))/(9009*b^4)

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Maple [A]  time = 0.007, size = 71, normalized size = 0.8 \[{\frac{1386\,{b}^{3}B{x}^{3}+1638\,A{x}^{2}{b}^{3}-756\,B{x}^{2}a{b}^{2}-728\,Axa{b}^{2}+336\,Bx{a}^{2}b+208\,A{a}^{2}b-96\,B{a}^{3}}{9009\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

2/9009*(b*x+a)^(7/2)*(693*B*b^3*x^3+819*A*b^3*x^2-378*B*a*b^2*x^2-364*A*a*b^2*x+
168*B*a^2*b*x+104*A*a^2*b-48*B*a^3)/b^4

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Maxima [A]  time = 1.35417, size = 104, normalized size = 1.09 \[ \frac{2 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} B - 819 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{11}{2}} + 1001 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{9}{2}} - 1287 \,{\left (B a^{3} - A a^{2} b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{9009 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^2,x, algorithm="maxima")

[Out]

2/9009*(693*(b*x + a)^(13/2)*B - 819*(3*B*a - A*b)*(b*x + a)^(11/2) + 1001*(3*B*
a^2 - 2*A*a*b)*(b*x + a)^(9/2) - 1287*(B*a^3 - A*a^2*b)*(b*x + a)^(7/2))/b^4

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Fricas [A]  time = 0.20383, size = 193, normalized size = 2.03 \[ \frac{2 \,{\left (693 \, B b^{6} x^{6} - 48 \, B a^{6} + 104 \, A a^{5} b + 63 \,{\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 7 \,{\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{4} +{\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{3} - 3 \,{\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt{b x + a}}{9009 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^2,x, algorithm="fricas")

[Out]

2/9009*(693*B*b^6*x^6 - 48*B*a^6 + 104*A*a^5*b + 63*(27*B*a*b^5 + 13*A*b^6)*x^5
+ 7*(159*B*a^2*b^4 + 299*A*a*b^5)*x^4 + (15*B*a^3*b^3 + 1469*A*a^2*b^4)*x^3 - 3*
(6*B*a^4*b^2 - 13*A*a^3*b^3)*x^2 + 4*(6*B*a^5*b - 13*A*a^4*b^2)*x)*sqrt(b*x + a)
/b^4

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Sympy [A]  time = 12.3751, size = 292, normalized size = 3.07 \[ \begin{cases} \frac{16 A a^{5} \sqrt{a + b x}}{693 b^{3}} - \frac{8 A a^{4} x \sqrt{a + b x}}{693 b^{2}} + \frac{2 A a^{3} x^{2} \sqrt{a + b x}}{231 b} + \frac{226 A a^{2} x^{3} \sqrt{a + b x}}{693} + \frac{46 A a b x^{4} \sqrt{a + b x}}{99} + \frac{2 A b^{2} x^{5} \sqrt{a + b x}}{11} - \frac{32 B a^{6} \sqrt{a + b x}}{3003 b^{4}} + \frac{16 B a^{5} x \sqrt{a + b x}}{3003 b^{3}} - \frac{4 B a^{4} x^{2} \sqrt{a + b x}}{1001 b^{2}} + \frac{10 B a^{3} x^{3} \sqrt{a + b x}}{3003 b} + \frac{106 B a^{2} x^{4} \sqrt{a + b x}}{429} + \frac{54 B a b x^{5} \sqrt{a + b x}}{143} + \frac{2 B b^{2} x^{6} \sqrt{a + b x}}{13} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{3}}{3} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

Piecewise((16*A*a**5*sqrt(a + b*x)/(693*b**3) - 8*A*a**4*x*sqrt(a + b*x)/(693*b*
*2) + 2*A*a**3*x**2*sqrt(a + b*x)/(231*b) + 226*A*a**2*x**3*sqrt(a + b*x)/693 +
46*A*a*b*x**4*sqrt(a + b*x)/99 + 2*A*b**2*x**5*sqrt(a + b*x)/11 - 32*B*a**6*sqrt
(a + b*x)/(3003*b**4) + 16*B*a**5*x*sqrt(a + b*x)/(3003*b**3) - 4*B*a**4*x**2*sq
rt(a + b*x)/(1001*b**2) + 10*B*a**3*x**3*sqrt(a + b*x)/(3003*b) + 106*B*a**2*x**
4*sqrt(a + b*x)/429 + 54*B*a*b*x**5*sqrt(a + b*x)/143 + 2*B*b**2*x**6*sqrt(a + b
*x)/13, Ne(b, 0)), (a**(5/2)*(A*x**3/3 + B*x**4/4), True))

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GIAC/XCAS [A]  time = 0.226312, size = 582, normalized size = 6.13 \[ \frac{2 \,{\left (\frac{429 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A a^{2}}{b^{14}} + \frac{143 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B a^{2}}{b^{27}} + \frac{286 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} A a}{b^{26}} + \frac{26 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} B a}{b^{43}} + \frac{13 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} A}{b^{42}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{60} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{60} + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{60} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{60} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{60} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{60}\right )} B}{b^{63}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^2,x, algorithm="giac")

[Out]

2/45045*(429*(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)
^(3/2)*a^2*b^12)*A*a^2/b^14 + 143*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)
*a*b^24 + 189*(b*x + a)^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*B*a^2/b^2
7 + 286*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 189*(b*x + a)^(5
/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*A*a/b^26 + 26*(315*(b*x + a)^(11/2)
*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 2772*(b*x
+ a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*B*a/b^43 + 13*(315*(b*x + a
)^(11/2)*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 27
72*(b*x + a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*A/b^42 + 5*(693*(b*
x + a)^(13/2)*b^60 - 4095*(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^(9/2)*a^2*b^
60 - 12870*(b*x + a)^(7/2)*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60 - 3003*(b*x
+ a)^(3/2)*a^5*b^60)*B/b^63)/b