3.500 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac{16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac{4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac{2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) + (2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(1
95*a^2*x^(13/2)) - (4*b*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(715*a^3*x^(11/2)) + (
16*b^2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(6435*a^4*x^(9/2)) - (32*b^3*(8*A*b - 1
5*a*B)*(a + b*x)^(7/2))/(45045*a^5*x^(7/2))

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Rubi [A]  time = 0.180177, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac{16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac{4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac{2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) + (2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(1
95*a^2*x^(13/2)) - (4*b*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(715*a^3*x^(11/2)) + (
16*b^2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(6435*a^4*x^(9/2)) - (32*b^3*(8*A*b - 1
5*a*B)*(a + b*x)^(7/2))/(45045*a^5*x^(7/2))

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Rubi in Sympy [A]  time = 15.0919, size = 150, normalized size = 1. \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{15 a x^{\frac{15}{2}}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (8 A b - 15 B a\right )}{195 a^{2} x^{\frac{13}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{7}{2}} \left (8 A b - 15 B a\right )}{715 a^{3} x^{\frac{11}{2}}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (8 A b - 15 B a\right )}{6435 a^{4} x^{\frac{9}{2}}} - \frac{32 b^{3} \left (a + b x\right )^{\frac{7}{2}} \left (8 A b - 15 B a\right )}{45045 a^{5} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(17/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(15*a*x**(15/2)) + 2*(a + b*x)**(7/2)*(8*A*b - 15*B*a)/(19
5*a**2*x**(13/2)) - 4*b*(a + b*x)**(7/2)*(8*A*b - 15*B*a)/(715*a**3*x**(11/2)) +
16*b**2*(a + b*x)**(7/2)*(8*A*b - 15*B*a)/(6435*a**4*x**(9/2)) - 32*b**3*(a + b
*x)**(7/2)*(8*A*b - 15*B*a)/(45045*a**5*x**(7/2))

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Mathematica [A]  time = 0.14474, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{7/2} \left (231 a^4 (13 A+15 B x)-42 a^3 b x (44 A+45 B x)+168 a^2 b^2 x^2 (6 A+5 B x)-16 a b^3 x^3 (28 A+15 B x)+128 A b^4 x^4\right )}{45045 a^5 x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(128*A*b^4*x^4 + 168*a^2*b^2*x^2*(6*A + 5*B*x) + 231*a^4*(13
*A + 15*B*x) - 16*a*b^3*x^3*(28*A + 15*B*x) - 42*a^3*b*x*(44*A + 45*B*x)))/(4504
5*a^5*x^(15/2))

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Maple [A]  time = 0.008, size = 101, normalized size = 0.7 \[ -{\frac{256\,A{b}^{4}{x}^{4}-480\,Ba{b}^{3}{x}^{4}-896\,Aa{b}^{3}{x}^{3}+1680\,B{a}^{2}{b}^{2}{x}^{3}+2016\,A{a}^{2}{b}^{2}{x}^{2}-3780\,B{a}^{3}b{x}^{2}-3696\,A{a}^{3}bx+6930\,B{a}^{4}x+6006\,A{a}^{4}}{45045\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*x^4-240*B*a*b^3*x^4-448*A*a*b^3*x^3+840*B*a^2*
b^2*x^3+1008*A*a^2*b^2*x^2-1890*B*a^3*b*x^2-1848*A*a^3*b*x+3465*B*a^4*x+3003*A*a
^4)/x^(15/2)/a^5

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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)^(5/2)/x^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235196, size = 234, normalized size = 1.56 \[ -\frac{2 \,{\left (3003 \, A a^{7} - 16 \,{\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 8 \,{\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{6} - 6 \,{\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{5} + 5 \,{\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{4} + 35 \,{\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 63 \,{\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{2} + 231 \,{\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x\right )} \sqrt{b x + a}}{45045 \, a^{5} x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*A*a^7 - 16*(15*B*a*b^6 - 8*A*b^7)*x^7 + 8*(15*B*a^2*b^5 - 8*A*a*b
^6)*x^6 - 6*(15*B*a^3*b^4 - 8*A*a^2*b^5)*x^5 + 5*(15*B*a^4*b^3 - 8*A*a^3*b^4)*x^
4 + 35*(159*B*a^5*b^2 + A*a^4*b^3)*x^3 + 63*(135*B*a^6*b + 71*A*a^5*b^2)*x^2 + 2
31*(15*B*a^7 + 31*A*a^6*b)*x)*sqrt(b*x + a)/(a^5*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)**(5/2)*(B*x+A)/x**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246039, size = 258, normalized size = 1.72 \[ -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (15 \, B a^{3} b^{14} - 8 \, A a^{2} b^{15}\right )}{\left (b x + a\right )}}{a^{8} b^{24}} - \frac{15 \,{\left (15 \, B a^{4} b^{14} - 8 \, A a^{3} b^{15}\right )}}{a^{8} b^{24}}\right )} + \frac{195 \,{\left (15 \, B a^{5} b^{14} - 8 \, A a^{4} b^{15}\right )}}{a^{8} b^{24}}\right )} - \frac{715 \,{\left (15 \, B a^{6} b^{14} - 8 \, A a^{5} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )} + \frac{6435 \,{\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )}^{\frac{7}{2}} b}{2952069120 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{15}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(17/2),x, algorithm="giac")

[Out]

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