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

Optimal. Leaf size=140 \[ \frac{3 c (4 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}-\frac{3 c \sqrt{x} (4 b B-5 A c)}{4 b^3 \sqrt{b x+c x^2}}-\frac{4 b B-5 A c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{A}{2 b x^{3/2} \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.275635, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{3 c (4 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}-\frac{3 c \sqrt{x} (4 b B-5 A c)}{4 b^3 \sqrt{b x+c x^2}}-\frac{4 b B-5 A c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{A}{2 b x^{3/2} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.4386, size = 133, normalized size = 0.95 \[ - \frac{A}{2 b x^{\frac{3}{2}} \sqrt{b x + c x^{2}}} + \frac{5 A c - 4 B b}{4 b^{2} \sqrt{x} \sqrt{b x + c x^{2}}} + \frac{3 c \sqrt{x} \left (5 A c - 4 B b\right )}{4 b^{3} \sqrt{b x + c x^{2}}} - \frac{3 c \left (5 A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(2*b*x**(3/2)*sqrt(b*x + c*x**2)) + (5*A*c - 4*B*b)/(4*b**2*sqrt(x)*sqrt(b*x
+ c*x**2)) + 3*c*sqrt(x)*(5*A*c - 4*B*b)/(4*b**3*sqrt(b*x + c*x**2)) - 3*c*(5*A*
c - 4*B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b**(7/2))

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Mathematica [A]  time = 0.162652, size = 105, normalized size = 0.75 \[ \frac{\sqrt{b} \left (A \left (-2 b^2+5 b c x+15 c^2 x^2\right )-4 b B x (b+3 c x)\right )+3 c x^2 \sqrt{b+c x} (4 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{4 b^{7/2} x^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*(-4*b*B*x*(b + 3*c*x) + A*(-2*b^2 + 5*b*c*x + 15*c^2*x^2)) + 3*c*(4*b*B
- 5*A*c)*x^2*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(4*b^(7/2)*x^(3/2)*S
qrt[x*(b + c*x)])

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Maple [A]  time = 0.033, size = 124, normalized size = 0.9 \[ -{\frac{1}{4\,cx+4\,b}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}{c}^{2}+4\,B{b}^{5/2}x+12\,B{b}^{3/2}{x}^{2}c-12\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}bc+2\,A{b}^{5/2}-5\,A{b}^{3/2}xc-15\,A\sqrt{b}{x}^{2}{c}^{2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/x^(5/2)*(x*(c*x+b))^(1/2)*(15*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2
)*x^2*c^2+4*B*b^(5/2)*x+12*B*b^(3/2)*x^2*c-12*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(
c*x+b)^(1/2)*x^2*b*c+2*A*b^(5/2)-5*A*b^(3/2)*x*c-15*A*b^(1/2)*x^2*c^2)/(c*x+b)/b
^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307047, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (2 \, A b^{2} + 3 \,{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} +{\left (4 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 3 \,{\left ({\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} +{\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{8 \,{\left (b^{3} c x^{4} + b^{4} x^{3}\right )} \sqrt{b}}, -\frac{{\left (2 \, A b^{2} + 3 \,{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} +{\left (4 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} - 3 \,{\left ({\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} +{\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{4 \,{\left (b^{3} c x^{4} + b^{4} x^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(2*(2*A*b^2 + 3*(4*B*b*c - 5*A*c^2)*x^2 + (4*B*b^2 - 5*A*b*c)*x)*sqrt(c*x^
2 + b*x)*sqrt(b)*sqrt(x) + 3*((4*B*b*c^2 - 5*A*c^3)*x^4 + (4*B*b^2*c - 5*A*b*c^2
)*x^3)*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2))/((b^3
*c*x^4 + b^4*x^3)*sqrt(b)), -1/4*((2*A*b^2 + 3*(4*B*b*c - 5*A*c^2)*x^2 + (4*B*b^
2 - 5*A*b*c)*x)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x) - 3*((4*B*b*c^2 - 5*A*c^3)*x^
4 + (4*B*b^2*c - 5*A*b*c^2)*x^3)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)))/((b
^3*c*x^4 + b^4*x^3)*sqrt(-b))]

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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**(3/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.369922, size = 169, normalized size = 1.21 \[ -\frac{3 \,{\left (4 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{4 \, \sqrt{-b} b^{3}} - \frac{2 \,{\left (B b c - A c^{2}\right )}}{\sqrt{c x + b} b^{3}} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c - 4 \, \sqrt{c x + b} B b^{2} c - 7 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{2} + 9 \, \sqrt{c x + b} A b c^{2}}{4 \, b^{3} c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(4*B*b*c - 5*A*c^2)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) - 2*(B*b*
c - A*c^2)/(sqrt(c*x + b)*b^3) - 1/4*(4*(c*x + b)^(3/2)*B*b*c - 4*sqrt(c*x + b)*
B*b^2*c - 7*(c*x + b)^(3/2)*A*c^2 + 9*sqrt(c*x + b)*A*b*c^2)/(b^3*c^2*x^2)