3.522 \(\int \frac{A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{32 b^2 \sqrt{a+b x} (8 A b-7 a B)}{35 a^5 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (8 A b-7 a B)}{35 a^4 x^{3/2}}+\frac{12 \sqrt{a+b x} (8 A b-7 a B)}{35 a^3 x^{5/2}}-\frac{2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt{a+b x}}-\frac{2 A}{7 a x^{7/2} \sqrt{a+b x}} \]

[Out]

(-2*A)/(7*a*x^(7/2)*Sqrt[a + b*x]) - (2*(8*A*b - 7*a*B))/(7*a^2*x^(5/2)*Sqrt[a +
b*x]) + (12*(8*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^3*x^(5/2)) - (16*b*(8*A*b - 7*
a*B)*Sqrt[a + b*x])/(35*a^4*x^(3/2)) + (32*b^2*(8*A*b - 7*a*B)*Sqrt[a + b*x])/(3
5*a^5*Sqrt[x])

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Rubi [A]  time = 0.170052, antiderivative size = 147, 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 \[ \frac{32 b^2 \sqrt{a+b x} (8 A b-7 a B)}{35 a^5 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (8 A b-7 a B)}{35 a^4 x^{3/2}}+\frac{12 \sqrt{a+b x} (8 A b-7 a B)}{35 a^3 x^{5/2}}-\frac{2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt{a+b x}}-\frac{2 A}{7 a x^{7/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*A)/(7*a*x^(7/2)*Sqrt[a + b*x]) - (2*(8*A*b - 7*a*B))/(7*a^2*x^(5/2)*Sqrt[a +
b*x]) + (12*(8*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^3*x^(5/2)) - (16*b*(8*A*b - 7*
a*B)*Sqrt[a + b*x])/(35*a^4*x^(3/2)) + (32*b^2*(8*A*b - 7*a*B)*Sqrt[a + b*x])/(3
5*a^5*Sqrt[x])

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Rubi in Sympy [A]  time = 14.9279, size = 144, normalized size = 0.98 \[ - \frac{2 A}{7 a x^{\frac{7}{2}} \sqrt{a + b x}} - \frac{2 \left (8 A b - 7 B a\right )}{7 a^{2} x^{\frac{5}{2}} \sqrt{a + b x}} + \frac{12 \sqrt{a + b x} \left (8 A b - 7 B a\right )}{35 a^{3} x^{\frac{5}{2}}} - \frac{16 b \sqrt{a + b x} \left (8 A b - 7 B a\right )}{35 a^{4} x^{\frac{3}{2}}} + \frac{32 b^{2} \sqrt{a + b x} \left (8 A b - 7 B a\right )}{35 a^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(9/2)/(b*x+a)**(3/2),x)

[Out]

-2*A/(7*a*x**(7/2)*sqrt(a + b*x)) - 2*(8*A*b - 7*B*a)/(7*a**2*x**(5/2)*sqrt(a +
b*x)) + 12*sqrt(a + b*x)*(8*A*b - 7*B*a)/(35*a**3*x**(5/2)) - 16*b*sqrt(a + b*x)
*(8*A*b - 7*B*a)/(35*a**4*x**(3/2)) + 32*b**2*sqrt(a + b*x)*(8*A*b - 7*B*a)/(35*
a**5*sqrt(x))

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Mathematica [A]  time = 0.111913, size = 94, normalized size = 0.64 \[ -\frac{2 \left (a^4 (5 A+7 B x)-2 a^3 b x (4 A+7 B x)+8 a^2 b^2 x^2 (2 A+7 B x)+16 a b^3 x^3 (7 B x-4 A)-128 A b^4 x^4\right )}{35 a^5 x^{7/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(-128*A*b^4*x^4 + 16*a*b^3*x^3*(-4*A + 7*B*x) + 8*a^2*b^2*x^2*(2*A + 7*B*x)
- 2*a^3*b*x*(4*A + 7*B*x) + a^4*(5*A + 7*B*x)))/(35*a^5*x^(7/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.01, size = 101, normalized size = 0.7 \[ -{\frac{-256\,A{b}^{4}{x}^{4}+224\,Ba{b}^{3}{x}^{4}-128\,Aa{b}^{3}{x}^{3}+112\,B{a}^{2}{b}^{2}{x}^{3}+32\,A{a}^{2}{b}^{2}{x}^{2}-28\,B{a}^{3}b{x}^{2}-16\,A{a}^{3}bx+14\,B{a}^{4}x+10\,A{a}^{4}}{35\,{a}^{5}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/(b*x+a)^(3/2),x)

[Out]

-2/35*(-128*A*b^4*x^4+112*B*a*b^3*x^4-64*A*a*b^3*x^3+56*B*a^2*b^2*x^3+16*A*a^2*b
^2*x^2-14*B*a^3*b*x^2-8*A*a^3*b*x+7*B*a^4*x+5*A*a^4)/x^(7/2)/(b*x+a)^(1/2)/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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23395, size = 136, normalized size = 0.93 \[ -\frac{2 \,{\left (5 \, A a^{4} + 16 \,{\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \,{\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 2 \,{\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} +{\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )}}{35 \, \sqrt{b x + a} a^{5} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(9/2)),x, algorithm="fricas")

[Out]

-2/35*(5*A*a^4 + 16*(7*B*a*b^3 - 8*A*b^4)*x^4 + 8*(7*B*a^2*b^2 - 8*A*a*b^3)*x^3
- 2*(7*B*a^3*b - 8*A*a^2*b^2)*x^2 + (7*B*a^4 - 8*A*a^3*b)*x)/(sqrt(b*x + a)*a^5*
x^(7/2))

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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**(9/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257309, size = 304, normalized size = 2.07 \[ \frac{{\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (77 \, B a^{10} b^{9}{\left | b \right |} - 93 \, A a^{9} b^{10}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{28 \,{\left (9 \, B a^{11} b^{9}{\left | b \right |} - 11 \, A a^{10} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} + \frac{70 \,{\left (4 \, B a^{12} b^{9}{\left | b \right |} - 5 \, A a^{11} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} - \frac{35 \,{\left (3 \, B a^{13} b^{9}{\left | b \right |} - 4 \, A a^{12} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} \sqrt{b x + a}}{26880 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}} - \frac{4 \,{\left (B a b^{\frac{9}{2}} - A b^{\frac{11}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{4}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(9/2)),x, algorithm="giac")

[Out]

1/26880*((b*x + a)*((b*x + a)*((77*B*a^10*b^9*abs(b) - 93*A*a^9*b^10*abs(b))*(b*
x + a)/(a^4*b^12) - 28*(9*B*a^11*b^9*abs(b) - 11*A*a^10*b^10*abs(b))/(a^4*b^12))
+ 70*(4*B*a^12*b^9*abs(b) - 5*A*a^11*b^10*abs(b))/(a^4*b^12)) - 35*(3*B*a^13*b^
9*abs(b) - 4*A*a^12*b^10*abs(b))/(a^4*b^12))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(
7/2) - 4*(B*a*b^(9/2) - A*b^(11/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2 + a*b)*a^4*abs(b))