3.523 \(\int \frac{A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=180 \[ -\frac{256 b^3 \sqrt{a+b x} (10 A b-9 a B)}{315 a^6 \sqrt{x}}+\frac{128 b^2 \sqrt{a+b x} (10 A b-9 a B)}{315 a^5 x^{3/2}}-\frac{32 b \sqrt{a+b x} (10 A b-9 a B)}{105 a^4 x^{5/2}}+\frac{16 \sqrt{a+b x} (10 A b-9 a B)}{63 a^3 x^{7/2}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}} \]

[Out]

(-2*A)/(9*a*x^(9/2)*Sqrt[a + b*x]) - (2*(10*A*b - 9*a*B))/(9*a^2*x^(7/2)*Sqrt[a
+ b*x]) + (16*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(63*a^3*x^(7/2)) - (32*b*(10*A*b -
9*a*B)*Sqrt[a + b*x])/(105*a^4*x^(5/2)) + (128*b^2*(10*A*b - 9*a*B)*Sqrt[a + b*
x])/(315*a^5*x^(3/2)) - (256*b^3*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(315*a^6*Sqrt[x
])

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Rubi [A]  time = 0.214812, antiderivative size = 180, 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 \[ -\frac{256 b^3 \sqrt{a+b x} (10 A b-9 a B)}{315 a^6 \sqrt{x}}+\frac{128 b^2 \sqrt{a+b x} (10 A b-9 a B)}{315 a^5 x^{3/2}}-\frac{32 b \sqrt{a+b x} (10 A b-9 a B)}{105 a^4 x^{5/2}}+\frac{16 \sqrt{a+b x} (10 A b-9 a B)}{63 a^3 x^{7/2}}-\frac{2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt{a+b x}}-\frac{2 A}{9 a x^{9/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(11/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*A)/(9*a*x^(9/2)*Sqrt[a + b*x]) - (2*(10*A*b - 9*a*B))/(9*a^2*x^(7/2)*Sqrt[a
+ b*x]) + (16*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(63*a^3*x^(7/2)) - (32*b*(10*A*b -
9*a*B)*Sqrt[a + b*x])/(105*a^4*x^(5/2)) + (128*b^2*(10*A*b - 9*a*B)*Sqrt[a + b*
x])/(315*a^5*x^(3/2)) - (256*b^3*(10*A*b - 9*a*B)*Sqrt[a + b*x])/(315*a^6*Sqrt[x
])

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Rubi in Sympy [A]  time = 19.5363, size = 178, normalized size = 0.99 \[ - \frac{2 A}{9 a x^{\frac{9}{2}} \sqrt{a + b x}} - \frac{2 \left (10 A b - 9 B a\right )}{9 a^{2} x^{\frac{7}{2}} \sqrt{a + b x}} + \frac{16 \sqrt{a + b x} \left (10 A b - 9 B a\right )}{63 a^{3} x^{\frac{7}{2}}} - \frac{32 b \sqrt{a + b x} \left (10 A b - 9 B a\right )}{105 a^{4} x^{\frac{5}{2}}} + \frac{128 b^{2} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{315 a^{5} x^{\frac{3}{2}}} - \frac{256 b^{3} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{315 a^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(11/2)/(b*x+a)**(3/2),x)

[Out]

-2*A/(9*a*x**(9/2)*sqrt(a + b*x)) - 2*(10*A*b - 9*B*a)/(9*a**2*x**(7/2)*sqrt(a +
b*x)) + 16*sqrt(a + b*x)*(10*A*b - 9*B*a)/(63*a**3*x**(7/2)) - 32*b*sqrt(a + b*
x)*(10*A*b - 9*B*a)/(105*a**4*x**(5/2)) + 128*b**2*sqrt(a + b*x)*(10*A*b - 9*B*a
)/(315*a**5*x**(3/2)) - 256*b**3*sqrt(a + b*x)*(10*A*b - 9*B*a)/(315*a**6*sqrt(x
))

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Mathematica [A]  time = 0.135493, size = 114, normalized size = 0.63 \[ -\frac{2 \left (5 a^5 (7 A+9 B x)-2 a^4 b x (25 A+36 B x)+16 a^3 b^2 x^2 (5 A+9 B x)-32 a^2 b^3 x^3 (5 A+18 B x)+128 a b^4 x^4 (5 A-9 B x)+1280 A b^5 x^5\right )}{315 a^6 x^{9/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(11/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(1280*A*b^5*x^5 + 128*a*b^4*x^4*(5*A - 9*B*x) + 16*a^3*b^2*x^2*(5*A + 9*B*x)
+ 5*a^5*(7*A + 9*B*x) - 32*a^2*b^3*x^3*(5*A + 18*B*x) - 2*a^4*b*x*(25*A + 36*B*
x)))/(315*a^6*x^(9/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.008, size = 125, normalized size = 0.7 \[ -{\frac{2560\,A{b}^{5}{x}^{5}-2304\,B{x}^{5}a{b}^{4}+1280\,aA{b}^{4}{x}^{4}-1152\,B{x}^{4}{a}^{2}{b}^{3}-320\,{a}^{2}A{b}^{3}{x}^{3}+288\,B{x}^{3}{a}^{3}{b}^{2}+160\,{a}^{3}A{b}^{2}{x}^{2}-144\,B{x}^{2}{a}^{4}b-100\,{a}^{4}Abx+90\,{a}^{5}Bx+70\,A{a}^{5}}{315\,{a}^{6}}{x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(11/2)/(b*x+a)^(3/2),x)

[Out]

-2/315*(1280*A*b^5*x^5-1152*B*a*b^4*x^5+640*A*a*b^4*x^4-576*B*a^2*b^3*x^4-160*A*
a^2*b^3*x^3+144*B*a^3*b^2*x^3+80*A*a^3*b^2*x^2-72*B*a^4*b*x^2-50*A*a^4*b*x+45*B*
a^5*x+35*A*a^5)/x^(9/2)/(b*x+a)^(1/2)/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^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23801, size = 170, normalized size = 0.94 \[ -\frac{2 \,{\left (35 \, A a^{5} - 128 \,{\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \,{\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 16 \,{\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \,{\left (9 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )}}{315 \, \sqrt{b x + a} a^{6} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(11/2)),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^5 - 128*(9*B*a*b^4 - 10*A*b^5)*x^5 - 64*(9*B*a^2*b^3 - 10*A*a*b^4
)*x^4 + 16*(9*B*a^3*b^2 - 10*A*a^2*b^3)*x^3 - 8*(9*B*a^4*b - 10*A*a^3*b^2)*x^2 +
5*(9*B*a^5 - 10*A*a^4*b)*x)/(sqrt(b*x + a)*a^6*x^(9/2))

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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)/x**(11/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269384, size = 328, normalized size = 1.82 \[ -\frac{{\left ({\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (837 \, B a^{15} b^{13} - 965 \, A a^{14} b^{14}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (401 \, B a^{16} b^{13} - 465 \, A a^{15} b^{14}\right )}}{a^{5} b^{15}}\right )} + \frac{126 \,{\left (47 \, B a^{17} b^{13} - 55 \, A a^{16} b^{14}\right )}}{a^{5} b^{15}}\right )} - \frac{210 \,{\left (21 \, B a^{18} b^{13} - 25 \, A a^{17} b^{14}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (4 \, B a^{19} b^{13} - 5 \, A a^{18} b^{14}\right )}}{a^{5} b^{15}}\right )} \sqrt{b x + a}}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}} + \frac{4 \,{\left (B a b^{\frac{11}{2}} - A b^{\frac{13}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{5}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(11/2)),x, algorithm="giac")

[Out]

-1/322560*(((b*x + a)*((b*x + a)*((837*B*a^15*b^13 - 965*A*a^14*b^14)*(b*x + a)/
(a^5*b^15) - 9*(401*B*a^16*b^13 - 465*A*a^15*b^14)/(a^5*b^15)) + 126*(47*B*a^17*
b^13 - 55*A*a^16*b^14)/(a^5*b^15)) - 210*(21*B*a^18*b^13 - 25*A*a^17*b^14)/(a^5*
b^15))*(b*x + a) + 315*(4*B*a^19*b^13 - 5*A*a^18*b^14)/(a^5*b^15))*sqrt(b*x + a)
/((b*x + a)*b - a*b)^(9/2) + 4*(B*a*b^(11/2) - A*b^(13/2))/(((sqrt(b*x + a)*sqrt
(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^5*abs(b))