3.550 \(\int \frac{(d+e x)^4}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=161 \[ \frac{\left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{5/2}}+\frac{e \sqrt{a+c x^2} \left (e x \left (26 c d^2-9 a e^2\right )+4 d \left (19 c d^2-16 a e^2\right )\right )}{24 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^3}{4 c}+\frac{7 d e \sqrt{a+c x^2} (d+e x)^2}{12 c} \]

[Out]

(7*d*e*(d + e*x)^2*Sqrt[a + c*x^2])/(12*c) + (e*(d + e*x)^3*Sqrt[a + c*x^2])/(4*
c) + (e*(4*d*(19*c*d^2 - 16*a*e^2) + e*(26*c*d^2 - 9*a*e^2)*x)*Sqrt[a + c*x^2])/
(24*c^2) + ((8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(8*c^(5/2))

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Rubi [A]  time = 0.452477, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{5/2}}+\frac{e \sqrt{a+c x^2} \left (e x \left (26 c d^2-9 a e^2\right )+4 d \left (19 c d^2-16 a e^2\right )\right )}{24 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^3}{4 c}+\frac{7 d e \sqrt{a+c x^2} (d+e x)^2}{12 c} \]

Antiderivative was successfully verified.

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

[Out]

(7*d*e*(d + e*x)^2*Sqrt[a + c*x^2])/(12*c) + (e*(d + e*x)^3*Sqrt[a + c*x^2])/(4*
c) + (e*(4*d*(19*c*d^2 - 16*a*e^2) + e*(26*c*d^2 - 9*a*e^2)*x)*Sqrt[a + c*x^2])/
(24*c^2) + ((8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 48.0877, size = 150, normalized size = 0.93 \[ \frac{7 d e \sqrt{a + c x^{2}} \left (d + e x\right )^{2}}{12 c} + \frac{e \sqrt{a + c x^{2}} \left (d + e x\right )^{3}}{4 c} - \frac{e \sqrt{a + c x^{2}} \left (d \left (64 a e^{2} - 76 c d^{2}\right ) + e x \left (9 a e^{2} - 26 c d^{2}\right )\right )}{24 c^{2}} + \frac{\left (3 a^{2} e^{4} - 24 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

7*d*e*sqrt(a + c*x**2)*(d + e*x)**2/(12*c) + e*sqrt(a + c*x**2)*(d + e*x)**3/(4*
c) - e*sqrt(a + c*x**2)*(d*(64*a*e**2 - 76*c*d**2) + e*x*(9*a*e**2 - 26*c*d**2))
/(24*c**2) + (3*a**2*e**4 - 24*a*c*d**2*e**2 + 8*c**2*d**4)*atanh(sqrt(c)*x/sqrt
(a + c*x**2))/(8*c**(5/2))

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Mathematica [A]  time = 0.166489, size = 126, normalized size = 0.78 \[ \frac{3 \left (3 a^2 e^4-24 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} e \sqrt{a+c x^2} \left (c \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )-a e^2 (64 d+9 e x)\right )}{24 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*e*Sqrt[a + c*x^2]*(-(a*e^2*(64*d + 9*e*x)) + c*(96*d^3 + 72*d^2*e*x + 3
2*d*e^2*x^2 + 6*e^3*x^3)) + 3*(8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^4)*Log[c*x +
Sqrt[c]*Sqrt[a + c*x^2]])/(24*c^(5/2))

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Maple [A]  time = 0.012, size = 198, normalized size = 1.2 \[{{d}^{4}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{4}{x}^{3}}{4\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,{e}^{4}ax}{8\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}{e}^{4}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+a}}-{\frac{8\,d{e}^{3}a}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}+3\,{\frac{{d}^{2}{e}^{2}x\sqrt{c{x}^{2}+a}}{c}}-3\,{\frac{{d}^{2}{e}^{2}a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}+4\,{\frac{{d}^{3}e\sqrt{c{x}^{2}+a}}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+a)^(1/2),x)

[Out]

d^4*ln(c^(1/2)*x+(c*x^2+a)^(1/2))/c^(1/2)+1/4*e^4*x^3/c*(c*x^2+a)^(1/2)-3/8*e^4*
a/c^2*x*(c*x^2+a)^(1/2)+3/8*e^4*a^2/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4/3*d*
e^3*x^2/c*(c*x^2+a)^(1/2)-8/3*d*e^3*a/c^2*(c*x^2+a)^(1/2)+3*d^2*e^2*x/c*(c*x^2+a
)^(1/2)-3*d^2*e^2*a/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4*d^3*e/c*(c*x^2+a)^(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/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238718, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (6 \, c e^{4} x^{3} + 32 \, c d e^{3} x^{2} + 96 \, c d^{3} e - 64 \, a d e^{3} + 9 \,{\left (8 \, c d^{2} e^{2} - a e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{48 \, c^{\frac{5}{2}}}, \frac{{\left (6 \, c e^{4} x^{3} + 32 \, c d e^{3} x^{2} + 96 \, c d^{3} e - 64 \, a d e^{3} + 9 \,{\left (8 \, c d^{2} e^{2} - a e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{24 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(6*c*e^4*x^3 + 32*c*d*e^3*x^2 + 96*c*d^3*e - 64*a*d*e^3 + 9*(8*c*d^2*e^
2 - a*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 3*(8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^
4)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/c^(5/2), 1/24*((6*c*e^4*
x^3 + 32*c*d*e^3*x^2 + 96*c*d^3*e - 64*a*d*e^3 + 9*(8*c*d^2*e^2 - a*e^4)*x)*sqrt
(c*x^2 + a)*sqrt(-c) + 3*(8*c^2*d^4 - 24*a*c*d^2*e^2 + 3*a^2*e^4)*arctan(sqrt(-c
)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 22.3865, size = 330, normalized size = 2.05 \[ - \frac{3 a^{\frac{3}{2}} e^{4} x}{8 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} d^{2} e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{c} - \frac{\sqrt{a} e^{4} x^{3}}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 a^{2} e^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{5}{2}}} - \frac{3 a d^{2} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{3}{2}}} + d^{4} \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + 4 d^{3} e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) + 4 d e^{3} \left (\begin{cases} - \frac{2 a \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{x^{2} \sqrt{a + c x^{2}}}{3 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) + \frac{e^{4} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

-3*a**(3/2)*e**4*x/(8*c**2*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*d**2*e**2*x*sqrt(1 +
c*x**2/a)/c - sqrt(a)*e**4*x**3/(8*c*sqrt(1 + c*x**2/a)) + 3*a**2*e**4*asinh(sqr
t(c)*x/sqrt(a))/(8*c**(5/2)) - 3*a*d**2*e**2*asinh(sqrt(c)*x/sqrt(a))/c**(3/2) +
d**4*Piecewise((sqrt(-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0) & (c < 0)), (sqr
t(a/c)*asinh(x*sqrt(c/a))/sqrt(a), (a > 0) & (c > 0)), (sqrt(-a/c)*acosh(x*sqrt(
-c/a))/sqrt(-a), (c > 0) & (a < 0))) + 4*d**3*e*Piecewise((x**2/(2*sqrt(a)), Eq(
c, 0)), (sqrt(a + c*x**2)/c, True)) + 4*d*e**3*Piecewise((-2*a*sqrt(a + c*x**2)/
(3*c**2) + x**2*sqrt(a + c*x**2)/(3*c), Ne(c, 0)), (x**4/(4*sqrt(a)), True)) + e
**4*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.220346, size = 180, normalized size = 1.12 \[ \frac{1}{24} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, x{\left (\frac{3 \, x e^{4}}{c} + \frac{16 \, d e^{3}}{c}\right )} + \frac{9 \,{\left (8 \, c^{3} d^{2} e^{2} - a c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac{32 \,{\left (3 \, c^{3} d^{3} e - 2 \, a c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac{{\left (8 \, c^{2} d^{4} - 24 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + a)*((2*x*(3*x*e^4/c + 16*d*e^3/c) + 9*(8*c^3*d^2*e^2 - a*c^2*e
^4)/c^4)*x + 32*(3*c^3*d^3*e - 2*a*c^2*d*e^3)/c^4) - 1/8*(8*c^2*d^4 - 24*a*c*d^2
*e^2 + 3*a^2*e^4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)