3.93 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^{15/2}} \, dx\)

Optimal. Leaf size=167 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{7/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}} \]

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(40*x^(9/2)) - (c^2*Sqrt[b*x + c*x^2])/(80*b*x^(7/2)) +
(c^3*Sqrt[b*x + c*x^2])/(64*b^2*x^(5/2)) - (3*c^4*Sqrt[b*x + c*x^2])/(128*b^3*x
^(3/2)) - (b*x + c*x^2)^(3/2)/(5*x^(13/2)) + (3*c^5*ArcTanh[Sqrt[b*x + c*x^2]/(S
qrt[b]*Sqrt[x])])/(128*b^(7/2))

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Rubi [A]  time = 0.233724, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{7/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^(15/2),x]

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(40*x^(9/2)) - (c^2*Sqrt[b*x + c*x^2])/(80*b*x^(7/2)) +
(c^3*Sqrt[b*x + c*x^2])/(64*b^2*x^(5/2)) - (3*c^4*Sqrt[b*x + c*x^2])/(128*b^3*x
^(3/2)) - (b*x + c*x^2)^(3/2)/(5*x^(13/2)) + (3*c^5*ArcTanh[Sqrt[b*x + c*x^2]/(S
qrt[b]*Sqrt[x])])/(128*b^(7/2))

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Rubi in Sympy [A]  time = 27.3097, size = 151, normalized size = 0.9 \[ - \frac{3 c \sqrt{b x + c x^{2}}}{40 x^{\frac{9}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 x^{\frac{13}{2}}} - \frac{c^{2} \sqrt{b x + c x^{2}}}{80 b x^{\frac{7}{2}}} + \frac{c^{3} \sqrt{b x + c x^{2}}}{64 b^{2} x^{\frac{5}{2}}} - \frac{3 c^{4} \sqrt{b x + c x^{2}}}{128 b^{3} x^{\frac{3}{2}}} + \frac{3 c^{5} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{128 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**(15/2),x)

[Out]

-3*c*sqrt(b*x + c*x**2)/(40*x**(9/2)) - (b*x + c*x**2)**(3/2)/(5*x**(13/2)) - c*
*2*sqrt(b*x + c*x**2)/(80*b*x**(7/2)) + c**3*sqrt(b*x + c*x**2)/(64*b**2*x**(5/2
)) - 3*c**4*sqrt(b*x + c*x**2)/(128*b**3*x**(3/2)) + 3*c**5*atanh(sqrt(b*x + c*x
**2)/(sqrt(b)*sqrt(x)))/(128*b**(7/2))

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Mathematica [A]  time = 0.208426, size = 116, normalized size = 0.69 \[ \frac{\sqrt{b+c x} \left (15 c^5 x^5 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} \left (128 b^4+176 b^3 c x+8 b^2 c^2 x^2-10 b c^3 x^3+15 c^4 x^4\right )\right )}{640 b^{7/2} x^{9/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^(15/2),x]

[Out]

(Sqrt[b + c*x]*(-(Sqrt[b]*Sqrt[b + c*x]*(128*b^4 + 176*b^3*c*x + 8*b^2*c^2*x^2 -
10*b*c^3*x^3 + 15*c^4*x^4)) + 15*c^5*x^5*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(640*
b^(7/2)*x^(9/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.027, size = 126, normalized size = 0.8 \[{\frac{1}{640}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{5}{x}^{5}-15\,{x}^{4}{c}^{4}\sqrt{b}\sqrt{cx+b}+10\,{x}^{3}{b}^{3/2}{c}^{3}\sqrt{cx+b}-8\,{x}^{2}{b}^{5/2}{c}^{2}\sqrt{cx+b}-176\,x{b}^{7/2}c\sqrt{cx+b}-128\,{b}^{9/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^(15/2),x)

[Out]

1/640*(x*(c*x+b))^(1/2)/b^(7/2)*(15*arctanh((c*x+b)^(1/2)/b^(1/2))*c^5*x^5-15*x^
4*c^4*b^(1/2)*(c*x+b)^(1/2)+10*x^3*b^(3/2)*c^3*(c*x+b)^(1/2)-8*x^2*b^(5/2)*c^2*(
c*x+b)^(1/2)-176*x*b^(7/2)*c*(c*x+b)^(1/2)-128*b^(9/2)*(c*x+b)^(1/2))/x^(11/2)/(
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((c*x^2 + b*x)^(3/2)/x^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252962, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, c^{5} x^{6} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (15 \, c^{4} x^{4} - 10 \, b c^{3} x^{3} + 8 \, b^{2} c^{2} x^{2} + 176 \, b^{3} c x + 128 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{1280 \, b^{\frac{7}{2}} x^{6}}, \frac{15 \, c^{5} x^{6} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (15 \, c^{4} x^{4} - 10 \, b c^{3} x^{3} + 8 \, b^{2} c^{2} x^{2} + 176 \, b^{3} c x + 128 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{640 \, \sqrt{-b} b^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(15/2),x, algorithm="fricas")

[Out]

[1/1280*(15*c^5*x^6*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) + (c*x^2 + 2*b*x)*sqrt(b
))/x^2) - 2*(15*c^4*x^4 - 10*b*c^3*x^3 + 8*b^2*c^2*x^2 + 176*b^3*c*x + 128*b^4)*
sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(7/2)*x^6), 1/640*(15*c^5*x^6*arctan(sqrt(
-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (15*c^4*x^4 - 10*b*c^3*x^3 + 8*b^2*c^2*x^2 + 17
6*b^3*c*x + 128*b^4)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3*x^6)]

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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((c*x**2+b*x)**(3/2)/x**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278703, size = 130, normalized size = 0.78 \[ -\frac{1}{640} \, c^{5}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x + b\right )}^{\frac{9}{2}} - 70 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 128 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3} - 15 \, \sqrt{c x + b} b^{4}}{b^{3} c^{5} x^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(15/2),x, algorithm="giac")

[Out]

-1/640*c^5*(15*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(c*x + b)^(9/
2) - 70*(c*x + b)^(7/2)*b + 128*(c*x + b)^(5/2)*b^2 + 70*(c*x + b)^(3/2)*b^3 - 1
5*sqrt(c*x + b)*b^4)/(b^3*c^5*x^5))