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

Optimal. Leaf size=207 \[ \frac{x \sqrt{a+c x^2} \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^2}+\frac{a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (3 e x \left (16 c d^2-5 a e^2\right )+8 d \left (13 c d^2-8 a e^2\right )\right )}{120 c^2}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^3}{6 c}+\frac{3 d e \left (a+c x^2\right )^{3/2} (d+e x)^2}{10 c} \]

[Out]

((8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*x*Sqrt[a + c*x^2])/(16*c^2) + (3*d*e*(d
+ e*x)^2*(a + c*x^2)^(3/2))/(10*c) + (e*(d + e*x)^3*(a + c*x^2)^(3/2))/(6*c) + (
e*(8*d*(13*c*d^2 - 8*a*e^2) + 3*e*(16*c*d^2 - 5*a*e^2)*x)*(a + c*x^2)^(3/2))/(12
0*c^2) + (a*(8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(16*c^(5/2))

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Rubi [A]  time = 0.60031, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{x \sqrt{a+c x^2} \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right )}{16 c^2}+\frac{a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (3 e x \left (16 c d^2-5 a e^2\right )+8 d \left (13 c d^2-8 a e^2\right )\right )}{120 c^2}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^3}{6 c}+\frac{3 d e \left (a+c x^2\right )^{3/2} (d+e x)^2}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

((8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*x*Sqrt[a + c*x^2])/(16*c^2) + (3*d*e*(d
+ e*x)^2*(a + c*x^2)^(3/2))/(10*c) + (e*(d + e*x)^3*(a + c*x^2)^(3/2))/(6*c) + (
e*(8*d*(13*c*d^2 - 8*a*e^2) + 3*e*(16*c*d^2 - 5*a*e^2)*x)*(a + c*x^2)^(3/2))/(12
0*c^2) + (a*(8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(16*c^(5/2))

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Rubi in Sympy [A]  time = 52.2617, size = 196, normalized size = 0.95 \[ \frac{a \left (a^{2} e^{4} - 12 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{5}{2}}} + \frac{3 d e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}{10 c} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{3}}{6 c} - \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d \left (192 a e^{2} - 312 c d^{2}\right ) + 9 e x \left (5 a e^{2} - 16 c d^{2}\right )\right )}{360 c^{2}} + \frac{x \sqrt{a + c x^{2}} \left (a^{2} e^{4} - 12 a c d^{2} e^{2} + 8 c^{2} d^{4}\right )}{16 c^{2}} \]

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]

a*(a**2*e**4 - 12*a*c*d**2*e**2 + 8*c**2*d**4)*atanh(sqrt(c)*x/sqrt(a + c*x**2))
/(16*c**(5/2)) + 3*d*e*(a + c*x**2)**(3/2)*(d + e*x)**2/(10*c) + e*(a + c*x**2)*
*(3/2)*(d + e*x)**3/(6*c) - e*(a + c*x**2)**(3/2)*(d*(192*a*e**2 - 312*c*d**2) +
9*e*x*(5*a*e**2 - 16*c*d**2))/(360*c**2) + x*sqrt(a + c*x**2)*(a**2*e**4 - 12*a
*c*d**2*e**2 + 8*c**2*d**4)/(16*c**2)

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Mathematica [A]  time = 0.217912, size = 177, normalized size = 0.86 \[ \frac{15 a \left (a^2 e^4-12 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} \sqrt{a+c x^2} \left (-a^2 e^3 (128 d+15 e x)+2 a c e \left (160 d^3+90 d^2 e x+32 d e^2 x^2+5 e^3 x^3\right )+8 c^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )}{240 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(-(a^2*e^3*(128*d + 15*e*x)) + 2*a*c*e*(160*d^3 + 90*d^
2*e*x + 32*d*e^2*x^2 + 5*e^3*x^3) + 8*c^2*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^
2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + 15*a*(8*c^2*d^4 - 12*a*c*d^2*e^2 + a^2*e^4)*Log
[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(240*c^(5/2))

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Maple [A]  time = 0.017, size = 260, normalized size = 1.3 \[{\frac{{d}^{4}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{4}a}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{4}{x}^{3}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}ax}{8\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{e}^{4}x}{16\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{{e}^{4}{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d{e}^{3}a}{15\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}{e}^{2}x}{2\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{2}{e}^{2}ax}{4\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{2}}{4}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}e}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^4*x*(c*x^2+a)^(1/2)+1/2*d^4*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/6*e^
4*x^3*(c*x^2+a)^(3/2)/c-1/8*e^4*a/c^2*x*(c*x^2+a)^(3/2)+1/16*e^4*a^2/c^2*x*(c*x^
2+a)^(1/2)+1/16*e^4*a^3/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4/5*d*e^3*x^2*(c*x
^2+a)^(3/2)/c-8/15*d*e^3*a/c^2*(c*x^2+a)^(3/2)+3/2*d^2*e^2*x*(c*x^2+a)^(3/2)/c-3
/4*d^2*e^2*a/c*x*(c*x^2+a)^(1/2)-3/4*d^2*e^2*a^2/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^
(1/2))+4/3*d^3*e*(c*x^2+a)^(3/2)/c

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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(sqrt(c*x^2 + a)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266781, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (40 \, c^{2} e^{4} x^{5} + 192 \, c^{2} d e^{3} x^{4} + 320 \, a c d^{3} e - 128 \, a^{2} d e^{3} + 10 \,{\left (36 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{3} + 64 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + 15 \,{\left (8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 15 \,{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{480 \, c^{\frac{5}{2}}}, \frac{{\left (40 \, c^{2} e^{4} x^{5} + 192 \, c^{2} d e^{3} x^{4} + 320 \, a c d^{3} e - 128 \, a^{2} d e^{3} + 10 \,{\left (36 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{3} + 64 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + 15 \,{\left (8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 15 \,{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{240 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/480*(2*(40*c^2*e^4*x^5 + 192*c^2*d*e^3*x^4 + 320*a*c*d^3*e - 128*a^2*d*e^3 +
10*(36*c^2*d^2*e^2 + a*c*e^4)*x^3 + 64*(5*c^2*d^3*e + a*c*d*e^3)*x^2 + 15*(8*c^2
*d^4 + 12*a*c*d^2*e^2 - a^2*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 15*(8*a*c^2*d^4 -
12*a^2*c*d^2*e^2 + a^3*e^4)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))
/c^(5/2), 1/240*((40*c^2*e^4*x^5 + 192*c^2*d*e^3*x^4 + 320*a*c*d^3*e - 128*a^2*d
*e^3 + 10*(36*c^2*d^2*e^2 + a*c*e^4)*x^3 + 64*(5*c^2*d^3*e + a*c*d*e^3)*x^2 + 15
*(8*c^2*d^4 + 12*a*c*d^2*e^2 - a^2*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) + 15*(8*a*c^
2*d^4 - 12*a^2*c*d^2*e^2 + a^3*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c
)*c^2)]

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Sympy [A]  time = 35.0645, size = 411, normalized size = 1.99 \[ - \frac{a^{\frac{5}{2}} e^{4} x}{16 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 a^{\frac{3}{2}} d^{2} e^{2} x}{4 c \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{\frac{3}{2}} e^{4} x^{3}}{48 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d^{4} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{9 \sqrt{a} d^{2} e^{2} x^{3}}{4 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 \sqrt{a} e^{4} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{3} e^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{5}{2}}} - \frac{3 a^{2} d^{2} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{4 c^{\frac{3}{2}}} + \frac{a d^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 4 d^{3} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 4 d e^{3} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{3 c d^{2} e^{2} x^{5}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c e^{4} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/2)*e**4*x/(16*c**2*sqrt(1 + c*x**2/a)) + 3*a**(3/2)*d**2*e**2*x/(4*c*sqrt
(1 + c*x**2/a)) - a**(3/2)*e**4*x**3/(48*c*sqrt(1 + c*x**2/a)) + sqrt(a)*d**4*x*
sqrt(1 + c*x**2/a)/2 + 9*sqrt(a)*d**2*e**2*x**3/(4*sqrt(1 + c*x**2/a)) + 5*sqrt(
a)*e**4*x**5/(24*sqrt(1 + c*x**2/a)) + a**3*e**4*asinh(sqrt(c)*x/sqrt(a))/(16*c*
*(5/2)) - 3*a**2*d**2*e**2*asinh(sqrt(c)*x/sqrt(a))/(4*c**(3/2)) + a*d**4*asinh(
sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + 4*d**3*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)),
((a + c*x**2)**(3/2)/(3*c), True)) + 4*d*e**3*Piecewise((-2*a**2*sqrt(a + c*x**2
)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)
), (sqrt(a)*x**4/4, True)) + 3*c*d**2*e**2*x**5/(2*sqrt(a)*sqrt(1 + c*x**2/a)) +
c*e**4*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.219254, size = 266, normalized size = 1.29 \[ \frac{1}{240} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x e^{4} + 24 \, d e^{3}\right )} x + \frac{5 \,{\left (36 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )}}{c^{4}}\right )} x + \frac{32 \,{\left (5 \, c^{4} d^{3} e + a c^{3} d e^{3}\right )}}{c^{4}}\right )} x + \frac{15 \,{\left (8 \, c^{4} d^{4} + 12 \, a c^{3} d^{2} e^{2} - a^{2} c^{2} e^{4}\right )}}{c^{4}}\right )} x + \frac{64 \,{\left (5 \, a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3}\right )}}{c^{4}}\right )} - \frac{{\left (8 \, a c^{2} d^{4} - 12 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*sqrt(c*x^2 + a)*((2*((4*(5*x*e^4 + 24*d*e^3)*x + 5*(36*c^4*d^2*e^2 + a*c^3
*e^4)/c^4)*x + 32*(5*c^4*d^3*e + a*c^3*d*e^3)/c^4)*x + 15*(8*c^4*d^4 + 12*a*c^3*
d^2*e^2 - a^2*c^2*e^4)/c^4)*x + 64*(5*a*c^3*d^3*e - 2*a^2*c^2*d*e^3)/c^4) - 1/16
*(8*a*c^2*d^4 - 12*a^2*c*d^2*e^2 + a^3*e^4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))
)/c^(5/2)