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

Optimal. Leaf size=84 \[ -\frac{4 b \sqrt{a+b x} (4 A b-5 a B)}{15 a^3 \sqrt{x}}+\frac{2 \sqrt{a+b x} (4 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A \sqrt{a+b x}}{5 a x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.103743, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{4 b \sqrt{a+b x} (4 A b-5 a B)}{15 a^3 \sqrt{x}}+\frac{2 \sqrt{a+b x} (4 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A \sqrt{a+b x}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.76138, size = 82, normalized size = 0.98 \[ - \frac{2 A \sqrt{a + b x}}{5 a x^{\frac{5}{2}}} + \frac{2 \sqrt{a + b x} \left (4 A b - 5 B a\right )}{15 a^{2} x^{\frac{3}{2}}} - \frac{4 b \sqrt{a + b x} \left (4 A b - 5 B a\right )}{15 a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0610483, size = 56, normalized size = 0.67 \[ -\frac{2 \sqrt{a+b x} \left (a^2 (3 A+5 B x)-2 a b x (2 A+5 B x)+8 A b^2 x^2\right )}{15 a^3 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 53, normalized size = 0.6 \[ -{\frac{16\,A{b}^{2}{x}^{2}-20\,B{x}^{2}ab-8\,aAbx+10\,{a}^{2}Bx+6\,A{a}^{2}}{15\,{a}^{3}}\sqrt{bx+a}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(b*x+a)^(1/2)*(8*A*b^2*x^2-10*B*a*b*x^2-4*A*a*b*x+5*B*a^2*x+3*A*a^2)/x^(5/
2)/a^3

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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 = 0.229253, size = 72, normalized size = 0.86 \[ -\frac{2 \,{\left (3 \, A a^{2} - 2 \,{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b x + a}}{15 \, a^{3} x^{\frac{5}{2}}} \]

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 - 2*(5*B*a*b - 4*A*b^2)*x^2 + (5*B*a^2 - 4*A*a*b)*x)*sqrt(b*x + a
)/(a^3*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221021, size = 154, normalized size = 1.83 \[ -\frac{\sqrt{b x + a}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (5 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )}}{a^{3} b^{9}}\right )} + \frac{15 \,{\left (B a^{3} b^{4} - A a^{2} b^{5}\right )}}{a^{3} b^{9}}\right )} b}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

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*sqrt(b*x + a)*((b*x + a)*(2*(5*B*a*b^4 - 4*A*b^5)*(b*x + a)/(a^3*b^9) - 5
*(5*B*a^2*b^4 - 4*A*a*b^5)/(a^3*b^9)) + 15*(B*a^3*b^4 - A*a^2*b^5)/(a^3*b^9))*b/
(((b*x + a)*b - a*b)^(5/2)*abs(b))