3.477 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{15/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{256 b^4 (a+b x)^{3/2} (10 A b-13 a B)}{45045 a^6 x^{3/2}}-\frac{128 b^3 (a+b x)^{3/2} (10 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{32 b^2 (a+b x)^{3/2} (10 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{16 b (a+b x)^{3/2} (10 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{3/2} (10 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(13*a*x^(13/2)) + (2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(
143*a^2*x^(11/2)) - (16*b*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(1287*a^3*x^(9/2))
+ (32*b^2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(3003*a^4*x^(7/2)) - (128*b^3*(10*A
*b - 13*a*B)*(a + b*x)^(3/2))/(15015*a^5*x^(5/2)) + (256*b^4*(10*A*b - 13*a*B)*(
a + b*x)^(3/2))/(45045*a^6*x^(3/2))

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Rubi [A]  time = 0.226252, 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 \[ \frac{256 b^4 (a+b x)^{3/2} (10 A b-13 a B)}{45045 a^6 x^{3/2}}-\frac{128 b^3 (a+b x)^{3/2} (10 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{32 b^2 (a+b x)^{3/2} (10 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{16 b (a+b x)^{3/2} (10 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{3/2} (10 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^(15/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(13*a*x^(13/2)) + (2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(
143*a^2*x^(11/2)) - (16*b*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(1287*a^3*x^(9/2))
+ (32*b^2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(3003*a^4*x^(7/2)) - (128*b^3*(10*A
*b - 13*a*B)*(a + b*x)^(3/2))/(15015*a^5*x^(5/2)) + (256*b^4*(10*A*b - 13*a*B)*(
a + b*x)^(3/2))/(45045*a^6*x^(3/2))

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Rubi in Sympy [A]  time = 19.5792, size = 184, normalized size = 1.01 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{13 a x^{\frac{13}{2}}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 13 B a\right )}{143 a^{2} x^{\frac{11}{2}}} - \frac{16 b \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 13 B a\right )}{1287 a^{3} x^{\frac{9}{2}}} + \frac{32 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 13 B a\right )}{3003 a^{4} x^{\frac{7}{2}}} - \frac{128 b^{3} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 13 B a\right )}{15015 a^{5} x^{\frac{5}{2}}} + \frac{256 b^{4} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 13 B a\right )}{45045 a^{6} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**(15/2),x)

[Out]

-2*A*(a + b*x)**(3/2)/(13*a*x**(13/2)) + 2*(a + b*x)**(3/2)*(10*A*b - 13*B*a)/(1
43*a**2*x**(11/2)) - 16*b*(a + b*x)**(3/2)*(10*A*b - 13*B*a)/(1287*a**3*x**(9/2)
) + 32*b**2*(a + b*x)**(3/2)*(10*A*b - 13*B*a)/(3003*a**4*x**(7/2)) - 128*b**3*(
a + b*x)**(3/2)*(10*A*b - 13*B*a)/(15015*a**5*x**(5/2)) + 256*b**4*(a + b*x)**(3
/2)*(10*A*b - 13*B*a)/(45045*a**6*x**(3/2))

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Mathematica [A]  time = 0.115984, size = 114, normalized size = 0.62 \[ -\frac{2 (a+b x)^{3/2} \left (315 a^5 (11 A+13 B x)-70 a^4 b x (45 A+52 B x)+80 a^3 b^2 x^2 (35 A+39 B x)-96 a^2 b^3 x^3 (25 A+26 B x)+128 a b^4 x^4 (15 A+13 B x)-1280 A b^5 x^5\right )}{45045 a^6 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(15/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(-1280*A*b^5*x^5 + 315*a^5*(11*A + 13*B*x) + 128*a*b^4*x^4*(
15*A + 13*B*x) - 96*a^2*b^3*x^3*(25*A + 26*B*x) + 80*a^3*b^2*x^2*(35*A + 39*B*x)
- 70*a^4*b*x*(45*A + 52*B*x)))/(45045*a^6*x^(13/2))

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Maple [A]  time = 0.008, size = 125, normalized size = 0.7 \[ -{\frac{-2560\,A{b}^{5}{x}^{5}+3328\,B{x}^{5}a{b}^{4}+3840\,aA{b}^{4}{x}^{4}-4992\,B{x}^{4}{a}^{2}{b}^{3}-4800\,{a}^{2}A{b}^{3}{x}^{3}+6240\,B{x}^{3}{a}^{3}{b}^{2}+5600\,{a}^{3}A{b}^{2}{x}^{2}-7280\,B{x}^{2}{a}^{4}b-6300\,{a}^{4}Abx+8190\,{a}^{5}Bx+6930\,A{a}^{5}}{45045\,{a}^{6}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^(15/2),x)

[Out]

-2/45045*(b*x+a)^(3/2)*(-1280*A*b^5*x^5+1664*B*a*b^4*x^5+1920*A*a*b^4*x^4-2496*B
*a^2*b^3*x^4-2400*A*a^2*b^3*x^3+3120*B*a^3*b^2*x^3+2800*A*a^3*b^2*x^2-3640*B*a^4
*b*x^2-3150*A*a^4*b*x+4095*B*a^5*x+3465*A*a^5)/x^(13/2)/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)*sqrt(b*x + a)/x^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238941, size = 201, normalized size = 1.1 \[ -\frac{2 \,{\left (3465 \, A a^{6} + 128 \,{\left (13 \, B a b^{5} - 10 \, A b^{6}\right )} x^{6} - 64 \,{\left (13 \, B a^{2} b^{4} - 10 \, A a b^{5}\right )} x^{5} + 48 \,{\left (13 \, B a^{3} b^{3} - 10 \, A a^{2} b^{4}\right )} x^{4} - 40 \,{\left (13 \, B a^{4} b^{2} - 10 \, A a^{3} b^{3}\right )} x^{3} + 35 \,{\left (13 \, B a^{5} b - 10 \, A a^{4} b^{2}\right )} x^{2} + 315 \,{\left (13 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt{b x + a}}{45045 \, a^{6} x^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(15/2),x, algorithm="fricas")

[Out]

-2/45045*(3465*A*a^6 + 128*(13*B*a*b^5 - 10*A*b^6)*x^6 - 64*(13*B*a^2*b^4 - 10*A
*a*b^5)*x^5 + 48*(13*B*a^3*b^3 - 10*A*a^2*b^4)*x^4 - 40*(13*B*a^4*b^2 - 10*A*a^3
*b^3)*x^3 + 35*(13*B*a^5*b - 10*A*a^4*b^2)*x^2 + 315*(13*B*a^6 + A*a^5*b)*x)*sqr
t(b*x + a)/(a^6*x^(13/2))

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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)*(b*x+a)**(1/2)/x**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239255, size = 296, normalized size = 1.62 \[ \frac{{\left ({\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a b^{12} - 10 \, A b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{2} b^{12} - 10 \, A a b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{3} b^{12} - 10 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{4} b^{12} - 10 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (13 \, B a^{5} b^{12} - 10 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} - \frac{15015 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} b}{33210777600 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(15/2),x, algorithm="giac")

[Out]

1/33210777600*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a*b^12 - 10*A*b^13)*(b*x +
a)/(a^7*b^21) - 13*(13*B*a^2*b^12 - 10*A*a*b^13)/(a^7*b^21)) + 143*(13*B*a^3*b^1
2 - 10*A*a^2*b^13)/(a^7*b^21)) - 429*(13*B*a^4*b^12 - 10*A*a^3*b^13)/(a^7*b^21))
*(b*x + a) + 3003*(13*B*a^5*b^12 - 10*A*a^4*b^13)/(a^7*b^21))*(b*x + a) - 15015*
(B*a^6*b^12 - A*a^5*b^13)/(a^7*b^21))*(b*x + a)^(3/2)*b/(((b*x + a)*b - a*b)^(13
/2)*abs(b))