3.226 \(\int \frac{x^{5/2} (A+B x)}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{16 b^2 \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt{x}}+\frac{8 b \sqrt{x} \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^3}-\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c} \]

[Out]

(-16*b^2*(6*b*B - 7*A*c)*Sqrt[b*x + c*x^2])/(105*c^4*Sqrt[x]) + (8*b*(6*b*B - 7*
A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(105*c^3) - (2*(6*b*B - 7*A*c)*x^(3/2)*Sqrt[b*x
+ c*x^2])/(35*c^2) + (2*B*x^(5/2)*Sqrt[b*x + c*x^2])/(7*c)

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Rubi [A]  time = 0.271582, antiderivative size = 133, 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 \[ -\frac{16 b^2 \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt{x}}+\frac{8 b \sqrt{x} \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^3}-\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/Sqrt[b*x + c*x^2],x]

[Out]

(-16*b^2*(6*b*B - 7*A*c)*Sqrt[b*x + c*x^2])/(105*c^4*Sqrt[x]) + (8*b*(6*b*B - 7*
A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(105*c^3) - (2*(6*b*B - 7*A*c)*x^(3/2)*Sqrt[b*x
+ c*x^2])/(35*c^2) + (2*B*x^(5/2)*Sqrt[b*x + c*x^2])/(7*c)

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Rubi in Sympy [A]  time = 16.3527, size = 129, normalized size = 0.97 \[ \frac{2 B x^{\frac{5}{2}} \sqrt{b x + c x^{2}}}{7 c} + \frac{16 b^{2} \left (7 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{105 c^{4} \sqrt{x}} - \frac{8 b \sqrt{x} \left (7 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{105 c^{3}} + \frac{2 x^{\frac{3}{2}} \left (7 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{35 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x)**(1/2),x)

[Out]

2*B*x**(5/2)*sqrt(b*x + c*x**2)/(7*c) + 16*b**2*(7*A*c - 6*B*b)*sqrt(b*x + c*x**
2)/(105*c**4*sqrt(x)) - 8*b*sqrt(x)*(7*A*c - 6*B*b)*sqrt(b*x + c*x**2)/(105*c**3
) + 2*x**(3/2)*(7*A*c - 6*B*b)*sqrt(b*x + c*x**2)/(35*c**2)

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Mathematica [A]  time = 0.0856025, size = 75, normalized size = 0.56 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^2 c (7 A+3 B x)-2 b c^2 x (14 A+9 B x)+3 c^3 x^2 (7 A+5 B x)-48 b^3 B\right )}{105 c^4 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-48*b^3*B + 8*b^2*c*(7*A + 3*B*x) + 3*c^3*x^2*(7*A + 5*B*x
) - 2*b*c^2*x*(14*A + 9*B*x)))/(105*c^4*Sqrt[x])

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Maple [A]  time = 0.009, size = 83, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 15\,B{c}^{3}{x}^{3}+21\,A{c}^{3}{x}^{2}-18\,Bb{c}^{2}{x}^{2}-28\,Ab{c}^{2}x+24\,B{b}^{2}cx+56\,A{b}^{2}c-48\,B{b}^{3} \right ) }{105\,{c}^{4}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(c*x^2+b*x)^(1/2),x)

[Out]

2/105*(c*x+b)*(15*B*c^3*x^3+21*A*c^3*x^2-18*B*b*c^2*x^2-28*A*b*c^2*x+24*B*b^2*c*
x+56*A*b^2*c-48*B*b^3)*x^(1/2)/c^4/(c*x^2+b*x)^(1/2)

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Maxima [A]  time = 0.70329, size = 132, normalized size = 0.99 \[ \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} A}{15 \, \sqrt{c x + b} c^{3}} + \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} B}{35 \, \sqrt{c x + b} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

2/15*(3*c^3*x^3 - b*c^2*x^2 + 4*b^2*c*x + 8*b^3)*A/(sqrt(c*x + b)*c^3) + 2/35*(5
*c^4*x^4 - b*c^3*x^3 + 2*b^2*c^2*x^2 - 8*b^3*c*x - 16*b^4)*B/(sqrt(c*x + b)*c^4)

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Fricas [A]  time = 0.288028, size = 144, normalized size = 1.08 \[ \frac{2 \,{\left (15 \, B c^{4} x^{5} - 3 \,{\left (B b c^{3} - 7 \, A c^{4}\right )} x^{4} +{\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 4 \,{\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} - 8 \,{\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )}}{105 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

2/105*(15*B*c^4*x^5 - 3*(B*b*c^3 - 7*A*c^4)*x^4 + (6*B*b^2*c^2 - 7*A*b*c^3)*x^3
- 4*(6*B*b^3*c - 7*A*b^2*c^2)*x^2 - 8*(6*B*b^4 - 7*A*b^3*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**(5/2)*(B*x+A)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274903, size = 149, normalized size = 1.12 \[ \frac{2 \,{\left (15 \,{\left (c x + b\right )}^{\frac{7}{2}} B - 63 \,{\left (c x + b\right )}^{\frac{5}{2}} B b + 105 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} - 105 \, \sqrt{c x + b} B b^{3} + 21 \,{\left (c x + b\right )}^{\frac{5}{2}} A c - 70 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c + 105 \, \sqrt{c x + b} A b^{2} c\right )}}{105 \, c^{4}} + \frac{16 \,{\left (6 \, B b^{\frac{7}{2}} - 7 \, A b^{\frac{5}{2}} c\right )}}{105 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

2/105*(15*(c*x + b)^(7/2)*B - 63*(c*x + b)^(5/2)*B*b + 105*(c*x + b)^(3/2)*B*b^2
- 105*sqrt(c*x + b)*B*b^3 + 21*(c*x + b)^(5/2)*A*c - 70*(c*x + b)^(3/2)*A*b*c +
105*sqrt(c*x + b)*A*b^2*c)/c^4 + 16/105*(6*B*b^(7/2) - 7*A*b^(5/2)*c)/c^4