3.210 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]

[Out]

(3*c*(4*b*B + A*c)*Sqrt[b*x + c*x^2])/(4*b*Sqrt[x]) - ((4*b*B + A*c)*(b*x + c*x^
2)^(3/2))/(4*b*x^(5/2)) - (A*(b*x + c*x^2)^(5/2))/(2*b*x^(9/2)) - (3*c*(4*b*B +
A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*Sqrt[b])

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Rubi [A]  time = 0.288395, antiderivative size = 137, 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 \[ \frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(9/2),x]

[Out]

(3*c*(4*b*B + A*c)*Sqrt[b*x + c*x^2])/(4*b*Sqrt[x]) - ((4*b*B + A*c)*(b*x + c*x^
2)^(3/2))/(4*b*x^(5/2)) - (A*(b*x + c*x^2)^(5/2))/(2*b*x^(9/2)) - (3*c*(4*b*B +
A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*Sqrt[b])

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Rubi in Sympy [A]  time = 17.7308, size = 124, normalized size = 0.91 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{2 b x^{\frac{9}{2}}} + \frac{3 c \left (A c + 4 B b\right ) \sqrt{b x + c x^{2}}}{4 b \sqrt{x}} - \frac{\left (A c + 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 b x^{\frac{5}{2}}} - \frac{3 c \left (A c + 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(9/2),x)

[Out]

-A*(b*x + c*x**2)**(5/2)/(2*b*x**(9/2)) + 3*c*(A*c + 4*B*b)*sqrt(b*x + c*x**2)/(
4*b*sqrt(x)) - (A*c + 4*B*b)*(b*x + c*x**2)**(3/2)/(4*b*x**(5/2)) - 3*c*(A*c + 4
*B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*sqrt(b))

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Mathematica [A]  time = 0.143693, size = 101, normalized size = 0.74 \[ -\frac{\sqrt{x (b+c x)} \left (3 c x^2 (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )+\sqrt{b} \sqrt{b+c x} (A (2 b+5 c x)+4 B x (b-2 c x))\right )}{4 \sqrt{b} x^{5/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(9/2),x]

[Out]

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

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Maple [A]  time = 0.026, size = 126, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}+12\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-8\,B{x}^{2}c\sqrt{b}\sqrt{cx+b}+5\,Axc\sqrt{cx+b}\sqrt{b}+4\,Bx{b}^{3/2}\sqrt{cx+b}+2\,A{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^(9/2),x)

[Out]

-1/4*(x*(c*x+b))^(1/2)*(3*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^2*c^2+12*B*arctanh(
(c*x+b)^(1/2)/b^(1/2))*x^2*b*c-8*B*x^2*c*b^(1/2)*(c*x+b)^(1/2)+5*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)/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)*(B*x + A)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299076, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{c x^{2} + b x} x^{\frac{3}{2}} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (8 \, B c^{2} x^{3} - 2 \, A b^{2} +{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} -{\left (4 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{b}}{8 \, \sqrt{c x^{2} + b x} \sqrt{b} x^{\frac{3}{2}}}, -\frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{c x^{2} + b x} x^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (8 \, B c^{2} x^{3} - 2 \, A b^{2} +{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} -{\left (4 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{-b}}{4 \, \sqrt{c x^{2} + b x} \sqrt{-b} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*(4*B*b*c + A*c^2)*sqrt(c*x^2 + b*x)*x^(3/2)*log((2*sqrt(c*x^2 + b*x)*b*s
qrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2) + 2*(8*B*c^2*x^3 - 2*A*b^2 + (4*B*b*c - 5
*A*c^2)*x^2 - (4*B*b^2 + 7*A*b*c)*x)*sqrt(b))/(sqrt(c*x^2 + b*x)*sqrt(b)*x^(3/2)
), -1/4*(3*(4*B*b*c + A*c^2)*sqrt(c*x^2 + b*x)*x^(3/2)*arctan(sqrt(-b)*sqrt(x)/s
qrt(c*x^2 + b*x)) - (8*B*c^2*x^3 - 2*A*b^2 + (4*B*b*c - 5*A*c^2)*x^2 - (4*B*b^2
+ 7*A*b*c)*x)*sqrt(-b))/(sqrt(c*x^2 + b*x)*sqrt(-b)*x^(3/2))]

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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)*(c*x**2+b*x)**(3/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313692, size = 161, normalized size = 1.18 \[ \frac{8 \, \sqrt{c x + b} B c^{2} + \frac{3 \,{\left (4 \, B b c^{2} + A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} - 3 \, \sqrt{c x + b} A b c^{3}}{c^{2} x^{2}}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(9/2),x, algorithm="giac")

[Out]

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