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

Optimal. Leaf size=75 \[ \frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-3 \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \]

[Out]

(3*c*Sqrt[b*x + c*x^2])/Sqrt[x] - (b*x + c*x^2)^(3/2)/x^(5/2) - 3*Sqrt[b]*c*ArcT
anh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])]

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Rubi [A]  time = 0.101937, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-3 \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*c*Sqrt[b*x + c*x^2])/Sqrt[x] - (b*x + c*x^2)^(3/2)/x^(5/2) - 3*Sqrt[b]*c*ArcT
anh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])]

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Rubi in Sympy [A]  time = 11.7504, size = 68, normalized size = 0.91 \[ - 3 \sqrt{b} c \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{3 c \sqrt{b x + c x^{2}}}{\sqrt{x}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0618703, size = 69, normalized size = 0.92 \[ -\frac{\sqrt{x (b+c x)} \left (\sqrt{b+c x} (b-2 c x)+3 \sqrt{b} c x \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{x^{3/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 68, normalized size = 0.9 \[{1 \left ( 2\,xc\sqrt{cx+b}\sqrt{b}-3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) xbc-{b}^{{\frac{3}{2}}}\sqrt{cx+b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*x*c*(c*x+b)^(1/2)*b^(1/2)-3*arctanh((c*x+b)^(1/2)/b^(1/2))*x*b*c-b^(3/2)*(c*x
+b)^(1/2))*(x*(c*x+b))^(1/2)/x^(3/2)/(c*x+b)^(1/2)/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((c*x^2 + b*x)^(3/2)/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235561, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c^{2} x^{2} + 3 \, \sqrt{c x^{2} + b x} \sqrt{b} c \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \, b c x - 2 \, b^{2}}{2 \, \sqrt{c x^{2} + b x} \sqrt{x}}, \frac{2 \, c^{2} x^{2} - 3 \, \sqrt{c x^{2} + b x} \sqrt{-b} c \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) + b c x - b^{2}}{\sqrt{c x^{2} + b x} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(4*c^2*x^2 + 3*sqrt(c*x^2 + b*x)*sqrt(b)*c*sqrt(x)*log(-(c*x^2 + 2*b*x - 2*
sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*b*c*x - 2*b^2)/(sqrt(c*x^2 + b*x)*sq
rt(x)), (2*c^2*x^2 - 3*sqrt(c*x^2 + b*x)*sqrt(-b)*c*sqrt(x)*arctan(b*sqrt(x)/(sq
rt(c*x^2 + b*x)*sqrt(-b))) + b*c*x - b^2)/(sqrt(c*x^2 + b*x)*sqrt(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**(7/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.234416, size = 68, normalized size = 0.91 \[{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{c x + b} - \frac{\sqrt{c x + b} b}{c x}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*b*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) + 2*sqrt(c*x + b) - sqrt(c*x + b)*b
/(c*x))*c