3.329 \(\int \frac{(d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ -\frac{2 (d+e x)^3 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{3 b^4 c^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{3 b^4 c \sqrt{b x+c x^2}}+\frac{2 e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

(-2*(d + e*x)^3*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (4*(d + e
*x)*(b*c*d^2*(4*c*d - 5*b*e) + (2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x
))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (2*e*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 3*b
^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^4*c^2) + (2*e^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c
*x^2]])/c^(5/2)

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Rubi [A]  time = 0.541648, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 (d+e x)^3 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{3 b^4 c^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{3 b^4 c \sqrt{b x+c x^2}}+\frac{2 e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^3*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (4*(d + e
*x)*(b*c*d^2*(4*c*d - 5*b*e) + (2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x
))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (2*e*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 3*b
^2*e^2)*Sqrt[b*x + c*x^2])/(3*b^4*c^2) + (2*e^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c
*x^2]])/c^(5/2)

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Rubi in Sympy [A]  time = 59.5357, size = 202, normalized size = 0.97 \[ \frac{2 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{2 \left (d + e x\right )^{3} \left (b d - x \left (b e - 2 c d\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (d + e x\right ) \left (- b c d^{2} \left (5 b e - 4 c d\right ) + x \left (b e - 2 c d\right ) \left (b^{2} e^{2} + 4 b c d e - 4 c^{2} d^{2}\right )\right )}{3 b^{4} c \sqrt{b x + c x^{2}}} - \frac{2 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (3 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )}{3 b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

2*e**4*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(5/2) - 2*(d + e*x)**3*(b*d - x*(b
*e - 2*c*d))/(3*b**2*(b*x + c*x**2)**(3/2)) + 4*(d + e*x)*(-b*c*d**2*(5*b*e - 4*
c*d) + x*(b*e - 2*c*d)*(b**2*e**2 + 4*b*c*d*e - 4*c**2*d**2))/(3*b**4*c*sqrt(b*x
+ c*x**2)) - 2*e*(b*e - 2*c*d)*sqrt(b*x + c*x**2)*(3*b**2*e**2 + 8*b*c*d*e - 8*
c**2*d**2)/(3*b**4*c**2)

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Mathematica [A]  time = 0.313931, size = 196, normalized size = 0.94 \[ \frac{x^{5/2} (b+c x)^3 \left (-\frac{8 \sqrt{x} (b e-c d)^3 (b e+2 c d)}{3 b^4 c^2 (b+c x)}-\frac{8 d^3 (3 b e-2 c d)}{3 b^4 \sqrt{x}}+\frac{2 \sqrt{x} (b e-c d)^4}{3 b^3 c^2 (b+c x)^2}-\frac{2 d^4}{3 b^3 x^{3/2}}\right )}{(x (b+c x))^{5/2}}+\frac{2 e^4 x^{5/2} (b+c x)^{5/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{5/2} (x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(b*x + c*x^2)^(5/2),x]

[Out]

(x^(5/2)*(b + c*x)^3*((-2*d^4)/(3*b^3*x^(3/2)) - (8*d^3*(-2*c*d + 3*b*e))/(3*b^4
*Sqrt[x]) + (2*(-(c*d) + b*e)^4*Sqrt[x])/(3*b^3*c^2*(b + c*x)^2) - (8*(-(c*d) +
b*e)^3*(2*c*d + b*e)*Sqrt[x])/(3*b^4*c^2*(b + c*x))))/(x*(b + c*x))^(5/2) + (2*e
^4*x^(5/2)*(b + c*x)^(5/2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(5/2)*(x*(
b + c*x))^(5/2))

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Maple [B]  time = 0.019, size = 447, normalized size = 2.2 \[ -{\frac{4\,{d}^{4}cx}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{d}^{4}}{3\,b} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{32\,{c}^{2}{d}^{4}x}{3\,{b}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{16\,{d}^{4}c}{3\,{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{{e}^{4}{x}^{3}}{3\,c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{{e}^{4}b{x}^{2}}{2\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{{e}^{4}{b}^{2}x}{6\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,{e}^{4}x}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{{e}^{4}b}{6\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{{e}^{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}-4\,{\frac{d{e}^{3}{x}^{2}}{c \left ( c{x}^{2}+bx \right ) ^{3/2}}}-{\frac{4\,d{e}^{3}bx}{3\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,d{e}^{3}x}{3\,bc}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{4\,d{e}^{3}}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-4\,{\frac{{d}^{2}{e}^{2}x}{c \left ( c{x}^{2}+bx \right ) ^{3/2}}}+8\,{\frac{{d}^{2}{e}^{2}x}{{b}^{2}\sqrt{c{x}^{2}+bx}}}+4\,{\frac{{d}^{2}{e}^{2}}{bc\sqrt{c{x}^{2}+bx}}}+{\frac{8\,{d}^{3}ex}{3\,b} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{64\,{d}^{3}ecx}{3\,{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{32\,{d}^{3}e}{3\,{b}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+b*x)^(5/2),x)

[Out]

-4/3*d^4/b^2/(c*x^2+b*x)^(3/2)*c*x-2/3*d^4/b/(c*x^2+b*x)^(3/2)+32/3*d^4*c^2/b^4/
(c*x^2+b*x)^(1/2)*x+16/3*d^4*c/b^3/(c*x^2+b*x)^(1/2)-1/3*e^4*x^3/c/(c*x^2+b*x)^(
3/2)+1/2*e^4*b/c^2*x^2/(c*x^2+b*x)^(3/2)+1/6*e^4*b^2/c^3/(c*x^2+b*x)^(3/2)*x-7/3
*e^4/c^2/(c*x^2+b*x)^(1/2)*x-1/6*e^4*b/c^3/(c*x^2+b*x)^(1/2)+e^4/c^(5/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-4*d*e^3*x^2/c/(c*x^2+b*x)^(3/2)-4/3*d*e^3*b/c
^2/(c*x^2+b*x)^(3/2)*x+8/3*d*e^3/b/c/(c*x^2+b*x)^(1/2)*x+4/3*d*e^3/c^2/(c*x^2+b*
x)^(1/2)-4*d^2*e^2/c/(c*x^2+b*x)^(3/2)*x+8*d^2*e^2/b^2/(c*x^2+b*x)^(1/2)*x+4*d^2
*e^2/b/c/(c*x^2+b*x)^(1/2)+8/3*d^3*e/b/(c*x^2+b*x)^(3/2)*x-64/3*d^3*e/b^3*c/(c*x
^2+b*x)^(1/2)*x-32/3*d^3*e/b^2/(c*x^2+b*x)^(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((e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236114, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (b^{4} c e^{4} x^{2} + b^{5} e^{4} x\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (b^{3} c^{2} d^{4} - 4 \,{\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{3} - 3 \,{\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - b^{5} e^{4}\right )} x^{2} - 6 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x\right )} \sqrt{c}}{3 \,{\left (b^{4} c^{3} x^{2} + b^{5} c^{2} x\right )} \sqrt{c x^{2} + b x} \sqrt{c}}, \frac{2 \,{\left (3 \,{\left (b^{4} c e^{4} x^{2} + b^{5} e^{4} x\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (b^{3} c^{2} d^{4} - 4 \,{\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{3} - 3 \,{\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - b^{5} e^{4}\right )} x^{2} - 6 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x\right )} \sqrt{-c}\right )}}{3 \,{\left (b^{4} c^{3} x^{2} + b^{5} c^{2} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(b^4*c*e^4*x^2 + b^5*e^4*x)*sqrt(c*x^2 + b*x)*log((2*c*x + b)*sqrt(c) +
2*sqrt(c*x^2 + b*x)*c) - 2*(b^3*c^2*d^4 - 4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c
^3*d^2*e^2 + b^3*c^2*d*e^3 - b^4*c*e^4)*x^3 - 3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e
+ 6*b^3*c^2*d^2*e^2 - b^5*e^4)*x^2 - 6*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)*x)*sqrt(c
))/((b^4*c^3*x^2 + b^5*c^2*x)*sqrt(c*x^2 + b*x)*sqrt(c)), 2/3*(3*(b^4*c*e^4*x^2
+ b^5*e^4*x)*sqrt(c*x^2 + b*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (b^3*c
^2*d^4 - 4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + b^3*c^2*d*e^3 - b^4*
c*e^4)*x^3 - 3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 - b^5*e^4)*x^
2 - 6*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)*x)*sqrt(-c))/((b^4*c^3*x^2 + b^5*c^2*x)*sq
rt(c*x^2 + b*x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d + e*x)**4/(x*(b + c*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.228218, size = 282, normalized size = 1.36 \[ -\frac{2 \,{\left (\frac{d^{4}}{b} -{\left (x{\left (\frac{4 \,{\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - b^{5} e^{4}\right )}}{b^{4} c^{2}}\right )} + \frac{6 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} - \frac{e^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="giac")

[Out]

-2/3*(d^4/b - (x*(4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + b^3*c^2*d*e
^3 - b^4*c*e^4)*x/(b^4*c^2) + 3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*
e^2 - b^5*e^4)/(b^4*c^2)) + 6*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)/(b^4*c^2))*x)/(c*x
^2 + b*x)^(3/2) - e^4*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^
(5/2)