3.234 \(\int \frac{x^{7/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=141 \[ \frac{16 b \sqrt{b x+c x^2} (6 b B-5 A c)}{15 c^4 \sqrt{x}}-\frac{8 \sqrt{x} \sqrt{b x+c x^2} (6 b B-5 A c)}{15 c^3}+\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-5 A c)}{5 b c^2}-\frac{2 x^{7/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

(-2*(b*B - A*c)*x^(7/2))/(b*c*Sqrt[b*x + c*x^2]) + (16*b*(6*b*B - 5*A*c)*Sqrt[b*
x + c*x^2])/(15*c^4*Sqrt[x]) - (8*(6*b*B - 5*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(15
*c^3) + (2*(6*b*B - 5*A*c)*x^(3/2)*Sqrt[b*x + c*x^2])/(5*b*c^2)

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Rubi [A]  time = 0.278662, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b \sqrt{b x+c x^2} (6 b B-5 A c)}{15 c^4 \sqrt{x}}-\frac{8 \sqrt{x} \sqrt{b x+c x^2} (6 b B-5 A c)}{15 c^3}+\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-5 A c)}{5 b c^2}-\frac{2 x^{7/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B - A*c)*x^(7/2))/(b*c*Sqrt[b*x + c*x^2]) + (16*b*(6*b*B - 5*A*c)*Sqrt[b*
x + c*x^2])/(15*c^4*Sqrt[x]) - (8*(6*b*B - 5*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(15
*c^3) + (2*(6*b*B - 5*A*c)*x^(3/2)*Sqrt[b*x + c*x^2])/(5*b*c^2)

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Rubi in Sympy [A]  time = 18.1963, size = 133, normalized size = 0.94 \[ - \frac{16 b \left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{4} \sqrt{x}} + \frac{8 \sqrt{x} \left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{3}} + \frac{2 x^{\frac{7}{2}} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{2 x^{\frac{3}{2}} \left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{5 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*b*(5*A*c - 6*B*b)*sqrt(b*x + c*x**2)/(15*c**4*sqrt(x)) + 8*sqrt(x)*(5*A*c -
6*B*b)*sqrt(b*x + c*x**2)/(15*c**3) + 2*x**(7/2)*(A*c - B*b)/(b*c*sqrt(b*x + c*x
**2)) - 2*x**(3/2)*(5*A*c - 6*B*b)*sqrt(b*x + c*x**2)/(5*b*c**2)

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Mathematica [A]  time = 0.0800082, size = 74, normalized size = 0.52 \[ \frac{2 \sqrt{x} \left (-8 b^2 c (5 A-3 B x)-2 b c^2 x (10 A+3 B x)+c^3 x^2 (5 A+3 B x)+48 b^3 B\right )}{15 c^4 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 83, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,B{c}^{3}{x}^{3}-5\,A{c}^{3}{x}^{2}+6\,Bb{c}^{2}{x}^{2}+20\,Ab{c}^{2}x-24\,B{b}^{2}cx+40\,A{b}^{2}c-48\,B{b}^{3} \right ) }{15\,{c}^{4}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.799328, size = 324, normalized size = 2.3 \[ \frac{2}{15} \, B{\left (\frac{{\left (3 \, c^{4} x^{3} - b c^{3} x^{2} + 4 \, b^{2} c^{2} x + 8 \, b^{3} c\right )} x^{3} - 2 \,{\left (b c^{3} x^{3} - 2 \, b^{2} c^{2} x^{2} - 7 \, b^{3} c x - 4 \, b^{4}\right )} x^{2} + 10 \,{\left (b^{2} c^{2} x^{3} + 2 \, b^{3} c x^{2} + b^{4} x\right )} x}{{\left (c^{5} x^{3} + b c^{4} x^{2}\right )} \sqrt{c x + b}} + \frac{30 \, b^{3}}{\sqrt{c x + b} c^{4}}\right )} + \frac{2}{3} \, A{\left (\frac{{\left (c^{3} x^{2} - b c^{2} x - 2 \, b^{2} c\right )} x^{2} - 2 \,{\left (b c^{2} x^{2} + 2 \, b^{2} c x + b^{3}\right )} x}{{\left (c^{4} x^{2} + b c^{3} x\right )} \sqrt{c x + b}} - \frac{6 \, b^{2}}{\sqrt{c x + b} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*(((3*c^4*x^3 - b*c^3*x^2 + 4*b^2*c^2*x + 8*b^3*c)*x^3 - 2*(b*c^3*x^3 - 2*
b^2*c^2*x^2 - 7*b^3*c*x - 4*b^4)*x^2 + 10*(b^2*c^2*x^3 + 2*b^3*c*x^2 + b^4*x)*x)
/((c^5*x^3 + b*c^4*x^2)*sqrt(c*x + b)) + 30*b^3/(sqrt(c*x + b)*c^4)) + 2/3*A*(((
c^3*x^2 - b*c^2*x - 2*b^2*c)*x^2 - 2*(b*c^2*x^2 + 2*b^2*c*x + b^3)*x)/((c^4*x^2
+ b*c^3*x)*sqrt(c*x + b)) - 6*b^2/(sqrt(c*x + b)*c^3))

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Fricas [A]  time = 0.285668, size = 115, normalized size = 0.82 \[ \frac{2 \,{\left (3 \, B c^{3} x^{4} -{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{3} + 4 \,{\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 8 \,{\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(x**(7/2)*(B*x+A)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276472, size = 146, normalized size = 1.04 \[ \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} B - 15 \,{\left (c x + b\right )}^{\frac{3}{2}} B b + 45 \, \sqrt{c x + b} B b^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c - 30 \, \sqrt{c x + b} A b c + \frac{15 \,{\left (B b^{3} - A b^{2} c\right )}}{\sqrt{c x + b}}\right )}}{15 \, c^{4}} - \frac{16 \,{\left (6 \, B b^{3} - 5 \, A b^{2} c\right )}}{15 \, \sqrt{b} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(c*x + b)^(5/2)*B - 15*(c*x + b)^(3/2)*B*b + 45*sqrt(c*x + b)*B*b^2 + 5*
(c*x + b)^(3/2)*A*c - 30*sqrt(c*x + b)*A*b*c + 15*(B*b^3 - A*b^2*c)/sqrt(c*x + b
))/c^4 - 16/15*(6*B*b^3 - 5*A*b^2*c)/(sqrt(b)*c^4)