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

Optimal. Leaf size=105 \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{\sqrt{b x+c x^2} (4 b B-A c)}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.218057, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{\sqrt{b x+c x^2} (4 b B-A c)}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.7225, size = 90, normalized size = 0.86 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 b x^{\frac{7}{2}}} + \frac{\left (A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{4 b x^{\frac{3}{2}}} + \frac{c \left (A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116279, size = 94, normalized size = 0.9 \[ -\frac{\sqrt{x (b+c x)} \left (c x^2 (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )+\sqrt{b} \sqrt{b+c x} (2 A b+A c x+4 b B x)\right )}{4 b^{3/2} x^{5/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 108, normalized size = 1. \[{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( A{\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ){x}^{2}{c}^{2}-4\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-Axc\sqrt{cx+b}\sqrt{b}-4\,Bx{b}^{3/2}\sqrt{cx+b}-2\,A{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(x*(c*x+b))^(1/2)/b^(3/2)*(A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^2*c^2-4*B*arct
anh((c*x+b)^(1/2)/b^(1/2))*x^2*b*c-A*x*c*(c*x+b)^(1/2)*b^(1/2)-4*B*x*b^(3/2)*(c*
x+b)^(1/2)-2*A*b^(3/2)*(c*x+b)^(1/2))/x^(5/2)/(c*x+b)^(1/2)

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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(sqrt(c*x^2 + b*x)*(B*x + A)/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301752, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B b c - A c^{2}\right )} x^{3} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, A b +{\left (4 \, B b + A c\right )} x\right )} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{3}{2}} x^{3}}, -\frac{{\left (4 \, B b c - A c^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, A b +{\left (4 \, B b + A c\right )} x\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30705, size = 149, normalized size = 1.42 \[ \frac{\frac{{\left (4 \, B b c^{2} - A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} +{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} + \sqrt{c x + b} A b c^{3}}{b c^{2} x^{2}}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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