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

Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0669468, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.32053, size = 49, normalized size = 0.92 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{5 a x^{\frac{5}{2}}} + \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (A b - \frac{5 B a}{2}\right )}{15 a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0478323, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{3/2} (3 a A+5 a B x-2 A b x)}{15 a^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 31, normalized size = 0.6 \[ -{\frac{-4\,Abx+10\,Bax+6\,Aa}{15\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^(7/2),x)

[Out]

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

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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^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229475, size = 69, normalized size = 1.3 \[ -\frac{2 \,{\left (3 \, A a^{2} +{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + A a b\right )} x\right )} \sqrt{b x + a}}{15 \, a^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3*A*a^2 + (5*B*a*b - 2*A*b^2)*x^2 + (5*B*a^2 + A*a*b)*x)*sqrt(b*x + a)/(a
^2*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)*(b*x+a)**(1/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236642, size = 107, normalized size = 2.02 \[ \frac{{\left (b x + a\right )}^{\frac{3}{2}} b{\left (\frac{{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )}}{a^{3} b^{9}}\right )}}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/960*(b*x + a)^(3/2)*b*((5*B*a*b^4 - 2*A*b^5)*(b*x + a)/(a^3*b^9) - 5*(B*a^2*b^
4 - A*a*b^5)/(a^3*b^9))/(((b*x + a)*b - a*b)^(5/2)*abs(b))