3.488 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{5/2} (8 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{16 b^2 (a+b x)^{5/2} (8 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{4 b (a+b x)^{5/2} (8 A b-13 a B)}{429 a^3 x^{9/2}}+\frac{2 (a+b x)^{5/2} (8 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}} \]

[Out]

(-2*A*(a + b*x)^(5/2))/(13*a*x^(13/2)) + (2*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(1
43*a^2*x^(11/2)) - (4*b*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(429*a^3*x^(9/2)) + (1
6*b^2*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(3003*a^4*x^(7/2)) - (32*b^3*(8*A*b - 13
*a*B)*(a + b*x)^(5/2))/(15015*a^5*x^(5/2))

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Rubi [A]  time = 0.176811, 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 \[ -\frac{32 b^3 (a+b x)^{5/2} (8 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{16 b^2 (a+b x)^{5/2} (8 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{4 b (a+b x)^{5/2} (8 A b-13 a B)}{429 a^3 x^{9/2}}+\frac{2 (a+b x)^{5/2} (8 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*A*(a + b*x)^(5/2))/(13*a*x^(13/2)) + (2*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(1
43*a^2*x^(11/2)) - (4*b*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(429*a^3*x^(9/2)) + (1
6*b^2*(8*A*b - 13*a*B)*(a + b*x)^(5/2))/(3003*a^4*x^(7/2)) - (32*b^3*(8*A*b - 13
*a*B)*(a + b*x)^(5/2))/(15015*a^5*x^(5/2))

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Rubi in Sympy [A]  time = 14.9635, size = 150, normalized size = 1. \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{13 a x^{\frac{13}{2}}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - 13 B a\right )}{143 a^{2} x^{\frac{11}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - 13 B a\right )}{429 a^{3} x^{\frac{9}{2}}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - 13 B a\right )}{3003 a^{4} x^{\frac{7}{2}}} - \frac{32 b^{3} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - 13 B a\right )}{15015 a^{5} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(15/2),x)

[Out]

-2*A*(a + b*x)**(5/2)/(13*a*x**(13/2)) + 2*(a + b*x)**(5/2)*(8*A*b - 13*B*a)/(14
3*a**2*x**(11/2)) - 4*b*(a + b*x)**(5/2)*(8*A*b - 13*B*a)/(429*a**3*x**(9/2)) +
16*b**2*(a + b*x)**(5/2)*(8*A*b - 13*B*a)/(3003*a**4*x**(7/2)) - 32*b**3*(a + b*
x)**(5/2)*(8*A*b - 13*B*a)/(15015*a**5*x**(5/2))

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Mathematica [A]  time = 0.118307, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{5/2} \left (105 a^4 (11 A+13 B x)-70 a^3 b x (12 A+13 B x)+40 a^2 b^2 x^2 (14 A+13 B x)-16 a b^3 x^3 (20 A+13 B x)+128 A b^4 x^4\right )}{15015 a^5 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*(a + b*x)^(5/2)*(128*A*b^4*x^4 + 105*a^4*(11*A + 13*B*x) - 70*a^3*b*x*(12*A
+ 13*B*x) + 40*a^2*b^2*x^2*(14*A + 13*B*x) - 16*a*b^3*x^3*(20*A + 13*B*x)))/(150
15*a^5*x^(13/2))

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Maple [A]  time = 0.009, size = 101, normalized size = 0.7 \[ -{\frac{256\,A{b}^{4}{x}^{4}-416\,Ba{b}^{3}{x}^{4}-640\,Aa{b}^{3}{x}^{3}+1040\,B{a}^{2}{b}^{2}{x}^{3}+1120\,A{a}^{2}{b}^{2}{x}^{2}-1820\,B{a}^{3}b{x}^{2}-1680\,A{a}^{3}bx+2730\,B{a}^{4}x+2310\,A{a}^{4}}{15015\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^(15/2),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(128*A*b^4*x^4-208*B*a*b^3*x^4-320*A*a*b^3*x^3+520*B*a^2*
b^2*x^3+560*A*a^2*b^2*x^2-910*B*a^3*b*x^2-840*A*a^3*b*x+1365*B*a^4*x+1155*A*a^4)
/x^(13/2)/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)^(3/2)/x^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229469, size = 201, normalized size = 1.34 \[ -\frac{2 \,{\left (1155 \, A a^{6} - 16 \,{\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{6} + 8 \,{\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{5} - 6 \,{\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{4} + 5 \,{\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{3} + 35 \,{\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 105 \,{\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{15015 \, a^{5} x^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(15/2),x, algorithm="fricas")

[Out]

-2/15015*(1155*A*a^6 - 16*(13*B*a*b^5 - 8*A*b^6)*x^6 + 8*(13*B*a^2*b^4 - 8*A*a*b
^5)*x^5 - 6*(13*B*a^3*b^3 - 8*A*a^2*b^4)*x^4 + 5*(13*B*a^4*b^2 - 8*A*a^3*b^3)*x^
3 + 35*(52*B*a^5*b + A*a^4*b^2)*x^2 + 105*(13*B*a^6 + 14*A*a^5*b)*x)*sqrt(b*x +
a)/(a^5*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)**(3/2)*(B*x+A)/x**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.249041, size = 255, normalized size = 1.7 \[ -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a^{2} b^{12} - 8 \, A a b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{3} b^{12} - 8 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{4} b^{12} - 8 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{5} b^{12} - 8 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{11070259200 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(15/2),x, algorithm="giac")

[Out]

-1/11070259200*((2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a^2*b^12 - 8*A*a*b^13)*(b*x +
a)/(a^7*b^21) - 13*(13*B*a^3*b^12 - 8*A*a^2*b^13)/(a^7*b^21)) + 143*(13*B*a^4*b
^12 - 8*A*a^3*b^13)/(a^7*b^21)) - 429*(13*B*a^5*b^12 - 8*A*a^4*b^13)/(a^7*b^21))
*(b*x + a) + 3003*(B*a^6*b^12 - A*a^5*b^13)/(a^7*b^21))*(b*x + a)^(5/2)*b/(((b*x
+ a)*b - a*b)^(13/2)*abs(b))