3.489 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{256 b^4 (a+b x)^{5/2} (2 A b-3 a B)}{45045 a^6 x^{5/2}}-\frac{128 b^3 (a+b x)^{5/2} (2 A b-3 a B)}{9009 a^5 x^{7/2}}+\frac{32 b^2 (a+b x)^{5/2} (2 A b-3 a B)}{1287 a^4 x^{9/2}}-\frac{16 b (a+b x)^{5/2} (2 A b-3 a B)}{429 a^3 x^{11/2}}+\frac{2 (a+b x)^{5/2} (2 A b-3 a B)}{39 a^2 x^{13/2}}-\frac{2 A (a+b x)^{5/2}}{15 a x^{15/2}} \]

[Out]

(-2*A*(a + b*x)^(5/2))/(15*a*x^(15/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(39
*a^2*x^(13/2)) - (16*b*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(429*a^3*x^(11/2)) + (32
*b^2*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(1287*a^4*x^(9/2)) - (128*b^3*(2*A*b - 3*a
*B)*(a + b*x)^(5/2))/(9009*a^5*x^(7/2)) + (256*b^4*(2*A*b - 3*a*B)*(a + b*x)^(5/
2))/(45045*a^6*x^(5/2))

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Rubi [A]  time = 0.214292, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{256 b^4 (a+b x)^{5/2} (2 A b-3 a B)}{45045 a^6 x^{5/2}}-\frac{128 b^3 (a+b x)^{5/2} (2 A b-3 a B)}{9009 a^5 x^{7/2}}+\frac{32 b^2 (a+b x)^{5/2} (2 A b-3 a B)}{1287 a^4 x^{9/2}}-\frac{16 b (a+b x)^{5/2} (2 A b-3 a B)}{429 a^3 x^{11/2}}+\frac{2 (a+b x)^{5/2} (2 A b-3 a B)}{39 a^2 x^{13/2}}-\frac{2 A (a+b x)^{5/2}}{15 a x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*A*(a + b*x)^(5/2))/(15*a*x^(15/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(39
*a^2*x^(13/2)) - (16*b*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(429*a^3*x^(11/2)) + (32
*b^2*(2*A*b - 3*a*B)*(a + b*x)^(5/2))/(1287*a^4*x^(9/2)) - (128*b^3*(2*A*b - 3*a
*B)*(a + b*x)^(5/2))/(9009*a^5*x^(7/2)) + (256*b^4*(2*A*b - 3*a*B)*(a + b*x)^(5/
2))/(45045*a^6*x^(5/2))

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Rubi in Sympy [A]  time = 20.0904, size = 184, normalized size = 1.01 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{15 a x^{\frac{15}{2}}} + \frac{4 \left (a + b x\right )^{\frac{5}{2}} \left (A b - \frac{3 B a}{2}\right )}{39 a^{2} x^{\frac{13}{2}}} - \frac{16 b \left (a + b x\right )^{\frac{5}{2}} \left (2 A b - 3 B a\right )}{429 a^{3} x^{\frac{11}{2}}} + \frac{64 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (A b - \frac{3 B a}{2}\right )}{1287 a^{4} x^{\frac{9}{2}}} - \frac{256 b^{3} \left (a + b x\right )^{\frac{5}{2}} \left (A b - \frac{3 B a}{2}\right )}{9009 a^{5} x^{\frac{7}{2}}} + \frac{512 b^{4} \left (a + b x\right )^{\frac{5}{2}} \left (A b - \frac{3 B a}{2}\right )}{45045 a^{6} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(17/2),x)

[Out]

-2*A*(a + b*x)**(5/2)/(15*a*x**(15/2)) + 4*(a + b*x)**(5/2)*(A*b - 3*B*a/2)/(39*
a**2*x**(13/2)) - 16*b*(a + b*x)**(5/2)*(2*A*b - 3*B*a)/(429*a**3*x**(11/2)) + 6
4*b**2*(a + b*x)**(5/2)*(A*b - 3*B*a/2)/(1287*a**4*x**(9/2)) - 256*b**3*(a + b*x
)**(5/2)*(A*b - 3*B*a/2)/(9009*a**5*x**(7/2)) + 512*b**4*(a + b*x)**(5/2)*(A*b -
3*B*a/2)/(45045*a**6*x**(5/2))

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Mathematica [A]  time = 0.145874, size = 111, normalized size = 0.61 \[ -\frac{2 (a+b x)^{5/2} \left (231 a^5 (13 A+15 B x)-210 a^4 b x (11 A+12 B x)+1680 a^3 b^2 x^2 (A+B x)-160 a^2 b^3 x^3 (7 A+6 B x)+128 a b^4 x^4 (5 A+3 B x)-256 A b^5 x^5\right )}{45045 a^6 x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*(a + b*x)^(5/2)*(-256*A*b^5*x^5 + 1680*a^3*b^2*x^2*(A + B*x) + 128*a*b^4*x^4
*(5*A + 3*B*x) - 160*a^2*b^3*x^3*(7*A + 6*B*x) - 210*a^4*b*x*(11*A + 12*B*x) + 2
31*a^5*(13*A + 15*B*x)))/(45045*a^6*x^(15/2))

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Maple [A]  time = 0.01, size = 125, normalized size = 0.7 \[ -{\frac{-512\,A{b}^{5}{x}^{5}+768\,B{x}^{5}a{b}^{4}+1280\,aA{b}^{4}{x}^{4}-1920\,B{x}^{4}{a}^{2}{b}^{3}-2240\,{a}^{2}A{b}^{3}{x}^{3}+3360\,B{x}^{3}{a}^{3}{b}^{2}+3360\,{a}^{3}A{b}^{2}{x}^{2}-5040\,B{x}^{2}{a}^{4}b-4620\,{a}^{4}Abx+6930\,{a}^{5}Bx+6006\,A{a}^{5}}{45045\,{a}^{6}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^(17/2),x)

[Out]

-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*x^5+384*B*a*b^4*x^5+640*A*a*b^4*x^4-960*B*a^2
*b^3*x^4-1120*A*a^2*b^3*x^3+1680*B*a^3*b^2*x^3+1680*A*a^3*b^2*x^2-2520*B*a^4*b*x
^2-2310*A*a^4*b*x+3465*B*a^5*x+3003*A*a^5)/x^(15/2)/a^6

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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*x + A)*(b*x + a)^(3/2)/x^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23084, size = 234, normalized size = 1.28 \[ -\frac{2 \,{\left (3003 \, A a^{7} + 128 \,{\left (3 \, B a b^{6} - 2 \, A b^{7}\right )} x^{7} - 64 \,{\left (3 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{6} + 48 \,{\left (3 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{5} - 40 \,{\left (3 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{4} + 35 \,{\left (3 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{3} + 63 \,{\left (70 \, B a^{6} b + A a^{5} b^{2}\right )} x^{2} + 231 \,{\left (15 \, B a^{7} + 16 \, A a^{6} b\right )} x\right )} \sqrt{b x + a}}{45045 \, a^{6} x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*A*a^7 + 128*(3*B*a*b^6 - 2*A*b^7)*x^7 - 64*(3*B*a^2*b^5 - 2*A*a*b
^6)*x^6 + 48*(3*B*a^3*b^4 - 2*A*a^2*b^5)*x^5 - 40*(3*B*a^4*b^3 - 2*A*a^3*b^4)*x^
4 + 35*(3*B*a^5*b^2 - 2*A*a^4*b^3)*x^3 + 63*(70*B*a^6*b + A*a^5*b^2)*x^2 + 231*(
15*B*a^7 + 16*A*a^6*b)*x)*sqrt(b*x + a)/(a^6*x^(15/2))

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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)**(3/2)*(B*x+A)/x**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.245918, size = 302, normalized size = 1.65 \[ \frac{{\left ({\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (3 \, B a^{2} b^{14} - 2 \, A a b^{15}\right )}{\left (b x + a\right )}}{a^{8} b^{24}} - \frac{15 \,{\left (3 \, B a^{3} b^{14} - 2 \, A a^{2} b^{15}\right )}}{a^{8} b^{24}}\right )} + \frac{195 \,{\left (3 \, B a^{4} b^{14} - 2 \, A a^{3} b^{15}\right )}}{a^{8} b^{24}}\right )} - \frac{715 \,{\left (3 \, B a^{5} b^{14} - 2 \, A a^{4} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )} + \frac{6435 \,{\left (3 \, B a^{6} b^{14} - 2 \, A a^{5} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )} - \frac{9009 \,{\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{2952069120 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{15}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(17/2),x, algorithm="giac")

[Out]

1/2952069120*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(3*B*a^2*b^14 - 2*A*a*b^15)*(b*x +
a)/(a^8*b^24) - 15*(3*B*a^3*b^14 - 2*A*a^2*b^15)/(a^8*b^24)) + 195*(3*B*a^4*b^1
4 - 2*A*a^3*b^15)/(a^8*b^24)) - 715*(3*B*a^5*b^14 - 2*A*a^4*b^15)/(a^8*b^24))*(b
*x + a) + 6435*(3*B*a^6*b^14 - 2*A*a^5*b^15)/(a^8*b^24))*(b*x + a) - 9009*(B*a^7
*b^14 - A*a^6*b^15)/(a^8*b^24))*(b*x + a)^(5/2)*b/(((b*x + a)*b - a*b)^(15/2)*ab
s(b))