3.521 \(\int \frac{A+B x}{x^{7/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{16 b \sqrt{a+b x} (6 A b-5 a B)}{15 a^4 \sqrt{x}}+\frac{8 \sqrt{a+b x} (6 A b-5 a B)}{15 a^3 x^{3/2}}-\frac{2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt{a+b x}}-\frac{2 A}{5 a x^{5/2} \sqrt{a+b x}} \]

[Out]

(-2*A)/(5*a*x^(5/2)*Sqrt[a + b*x]) - (2*(6*A*b - 5*a*B))/(5*a^2*x^(3/2)*Sqrt[a +
b*x]) + (8*(6*A*b - 5*a*B)*Sqrt[a + b*x])/(15*a^3*x^(3/2)) - (16*b*(6*A*b - 5*a
*B)*Sqrt[a + b*x])/(15*a^4*Sqrt[x])

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Rubi [A]  time = 0.132678, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{16 b \sqrt{a+b x} (6 A b-5 a B)}{15 a^4 \sqrt{x}}+\frac{8 \sqrt{a+b x} (6 A b-5 a B)}{15 a^3 x^{3/2}}-\frac{2 (6 A b-5 a B)}{5 a^2 x^{3/2} \sqrt{a+b x}}-\frac{2 A}{5 a x^{5/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*A)/(5*a*x^(5/2)*Sqrt[a + b*x]) - (2*(6*A*b - 5*a*B))/(5*a^2*x^(3/2)*Sqrt[a +
b*x]) + (8*(6*A*b - 5*a*B)*Sqrt[a + b*x])/(15*a^3*x^(3/2)) - (16*b*(6*A*b - 5*a
*B)*Sqrt[a + b*x])/(15*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 10.8783, size = 110, normalized size = 0.96 \[ - \frac{2 A}{5 a x^{\frac{5}{2}} \sqrt{a + b x}} - \frac{2 \left (6 A b - 5 B a\right )}{5 a^{2} x^{\frac{3}{2}} \sqrt{a + b x}} + \frac{8 \sqrt{a + b x} \left (6 A b - 5 B a\right )}{15 a^{3} x^{\frac{3}{2}}} - \frac{16 b \sqrt{a + b x} \left (6 A b - 5 B a\right )}{15 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(b*x+a)**(3/2),x)

[Out]

-2*A/(5*a*x**(5/2)*sqrt(a + b*x)) - 2*(6*A*b - 5*B*a)/(5*a**2*x**(3/2)*sqrt(a +
b*x)) + 8*sqrt(a + b*x)*(6*A*b - 5*B*a)/(15*a**3*x**(3/2)) - 16*b*sqrt(a + b*x)*
(6*A*b - 5*B*a)/(15*a**4*sqrt(x))

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Mathematica [A]  time = 0.0888394, size = 75, normalized size = 0.66 \[ -\frac{2 \left (a^3 (3 A+5 B x)-2 a^2 b x (3 A+10 B x)+8 a b^2 x^2 (3 A-5 B x)+48 A b^3 x^3\right )}{15 a^4 x^{5/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(48*A*b^3*x^3 + 8*a*b^2*x^2*(3*A - 5*B*x) + a^3*(3*A + 5*B*x) - 2*a^2*b*x*(3
*A + 10*B*x)))/(15*a^4*x^(5/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.008, size = 77, normalized size = 0.7 \[ -{\frac{96\,A{b}^{3}{x}^{3}-80\,B{x}^{3}a{b}^{2}+48\,aA{b}^{2}{x}^{2}-40\,B{x}^{2}{a}^{2}b-12\,{a}^{2}Abx+10\,{a}^{3}Bx+6\,A{a}^{3}}{15\,{a}^{4}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b*x+a)^(3/2),x)

[Out]

-2/15*(48*A*b^3*x^3-40*B*a*b^2*x^3+24*A*a*b^2*x^2-20*B*a^2*b*x^2-6*A*a^2*b*x+5*B
*a^3*x+3*A*a^3)/x^(5/2)/(b*x+a)^(1/2)/a^4

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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^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237898, size = 104, normalized size = 0.91 \[ -\frac{2 \,{\left (3 \, A a^{3} - 8 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} +{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )}}{15 \, \sqrt{b x + a} a^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(7/2)),x, algorithm="fricas")

[Out]

-2/15*(3*A*a^3 - 8*(5*B*a*b^2 - 6*A*b^3)*x^3 - 4*(5*B*a^2*b - 6*A*a*b^2)*x^2 + (
5*B*a^3 - 6*A*a^2*b)*x)/(sqrt(b*x + a)*a^4*x^(5/2))

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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**(7/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250521, size = 236, normalized size = 2.07 \[ -\frac{\sqrt{b x + a}{\left ({\left (b x + a\right )}{\left (\frac{{\left (25 \, B a^{6} b^{7} - 33 \, A a^{5} b^{8}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (11 \, B a^{7} b^{7} - 15 \, A a^{6} b^{8}\right )}}{a^{3} b^{9}}\right )} + \frac{15 \,{\left (2 \, B a^{8} b^{7} - 3 \, A a^{7} b^{8}\right )}}{a^{3} b^{9}}\right )}}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}} + \frac{4 \,{\left (B a b^{\frac{7}{2}} - A b^{\frac{9}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(7/2)),x, algorithm="giac")

[Out]

-1/960*sqrt(b*x + a)*((b*x + a)*((25*B*a^6*b^7 - 33*A*a^5*b^8)*(b*x + a)/(a^3*b^
9) - 5*(11*B*a^7*b^7 - 15*A*a^6*b^8)/(a^3*b^9)) + 15*(2*B*a^8*b^7 - 3*A*a^7*b^8)
/(a^3*b^9))/((b*x + a)*b - a*b)^(5/2) + 4*(B*a*b^(7/2) - A*b^(9/2))/(((sqrt(b*x
+ a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^3*abs(b))