3.208 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ -2 A b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 A b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]

[Out]

(2*A*b*Sqrt[b*x + c*x^2])/Sqrt[x] + (2*A*(b*x + c*x^2)^(3/2))/(3*x^(3/2)) + (2*B
*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2)) - 2*A*b^(3/2)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqr
t[b]*Sqrt[x])]

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Rubi [A]  time = 0.216777, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -2 A b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 A b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(5/2),x]

[Out]

(2*A*b*Sqrt[b*x + c*x^2])/Sqrt[x] + (2*A*(b*x + c*x^2)^(3/2))/(3*x^(3/2)) + (2*B
*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2)) - 2*A*b^(3/2)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqr
t[b]*Sqrt[x])]

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Rubi in Sympy [A]  time = 15.7136, size = 99, normalized size = 0.94 \[ - 2 A b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 A b \sqrt{b x + c x^{2}}}{\sqrt{x}} + \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} + \frac{2 B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 c x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(5/2),x)

[Out]

-2*A*b**(3/2)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x))) + 2*A*b*sqrt(b*x + c*x
**2)/sqrt(x) + 2*A*(b*x + c*x**2)**(3/2)/(3*x**(3/2)) + 2*B*(b*x + c*x**2)**(5/2
)/(5*c*x**(5/2))

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Mathematica [A]  time = 0.17198, size = 100, normalized size = 0.95 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} \left (b (20 A c+6 B c x)+c^2 x (5 A+3 B x)+3 b^2 B\right )-15 A b^{3/2} c \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{15 c \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(5/2),x]

[Out]

(2*Sqrt[x]*Sqrt[b + c*x]*(Sqrt[b + c*x]*(3*b^2*B + c^2*x*(5*A + 3*B*x) + b*(20*A
*c + 6*B*c*x)) - 15*A*b^(3/2)*c*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(15*c*Sqrt[x*(b
+ c*x)])

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Maple [A]  time = 0.017, size = 113, normalized size = 1.1 \[ -{\frac{2}{15\,c}\sqrt{x \left ( cx+b \right ) } \left ( -3\,B{x}^{2}{c}^{2}\sqrt{cx+b}+15\,A{b}^{3/2}c{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -5\,Ax{c}^{2}\sqrt{cx+b}-6\,Bxbc\sqrt{cx+b}-20\,A\sqrt{cx+b}bc-3\,B{b}^{2}\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^(5/2),x)

[Out]

-2/15*(x*(c*x+b))^(1/2)*(-3*B*x^2*c^2*(c*x+b)^(1/2)+15*A*b^(3/2)*c*arctanh((c*x+
b)^(1/2)/b^(1/2))-5*A*x*c^2*(c*x+b)^(1/2)-6*B*x*b*c*(c*x+b)^(1/2)-20*A*(c*x+b)^(
1/2)*b*c-3*B*b^2*(c*x+b)^(1/2))/x^(1/2)/(c*x+b)^(1/2)/c

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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((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281814, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, B c^{3} x^{4} + 15 \, \sqrt{c x^{2} + b x} A b^{\frac{3}{2}} c \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (9 \, B b c^{2} + 5 \, A c^{3}\right )} x^{3} + 2 \,{\left (9 \, B b^{2} c + 25 \, A b c^{2}\right )} x^{2} + 2 \,{\left (3 \, B b^{3} + 20 \, A b^{2} c\right )} x}{15 \, \sqrt{c x^{2} + b x} c \sqrt{x}}, \frac{2 \,{\left (3 \, B c^{3} x^{4} - 15 \, \sqrt{c x^{2} + b x} A \sqrt{-b} b c \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) +{\left (9 \, B b c^{2} + 5 \, A c^{3}\right )} x^{3} +{\left (9 \, B b^{2} c + 25 \, A b c^{2}\right )} x^{2} +{\left (3 \, B b^{3} + 20 \, A b^{2} c\right )} x\right )}}{15 \, \sqrt{c x^{2} + b x} c \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(5/2),x, algorithm="fricas")

[Out]

[1/15*(6*B*c^3*x^4 + 15*sqrt(c*x^2 + b*x)*A*b^(3/2)*c*sqrt(x)*log(-(c*x^2 + 2*b*
x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*(9*B*b*c^2 + 5*A*c^3)*x^3 + 2*
(9*B*b^2*c + 25*A*b*c^2)*x^2 + 2*(3*B*b^3 + 20*A*b^2*c)*x)/(sqrt(c*x^2 + b*x)*c*
sqrt(x)), 2/15*(3*B*c^3*x^4 - 15*sqrt(c*x^2 + b*x)*A*sqrt(-b)*b*c*sqrt(x)*arctan
(b*sqrt(x)/(sqrt(c*x^2 + b*x)*sqrt(-b))) + (9*B*b*c^2 + 5*A*c^3)*x^3 + (9*B*b^2*
c + 25*A*b*c^2)*x^2 + (3*B*b^3 + 20*A*b^2*c)*x)/(sqrt(c*x^2 + b*x)*c*sqrt(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(5/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.28089, size = 166, normalized size = 1.58 \[ \frac{2 \, A b^{2} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \,{\left (15 \, A b^{2} c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 3 \, B \sqrt{-b} b^{\frac{5}{2}} + 20 \, A \sqrt{-b} b^{\frac{3}{2}} c\right )}}{15 \, \sqrt{-b} c} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} B c^{4} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{5} + 15 \, \sqrt{c x + b} A b c^{5}\right )}}{15 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(5/2),x, algorithm="giac")

[Out]

2*A*b^2*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - 2/15*(15*A*b^2*c*arctan(sqrt(b
)/sqrt(-b)) + 3*B*sqrt(-b)*b^(5/2) + 20*A*sqrt(-b)*b^(3/2)*c)/(sqrt(-b)*c) + 2/1
5*(3*(c*x + b)^(5/2)*B*c^4 + 5*(c*x + b)^(3/2)*A*c^5 + 15*sqrt(c*x + b)*A*b*c^5)
/c^5