3.523 \(\int (d+e x)^4 \left (a+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{x \left (a+c x^2\right )^{3/2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{64 c^2}+\frac{3 a x \sqrt{a+c x^2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{128 c^2}+\frac{3 a^2 \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (5 e x \left (26 c d^2-7 a e^2\right )+4 d \left (67 c d^2-32 a e^2\right )\right )}{560 c^2}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^3}{8 c}+\frac{11 d e \left (a+c x^2\right )^{5/2} (d+e x)^2}{56 c} \]

[Out]

(3*a*(16*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*x*Sqrt[a + c*x^2])/(128*c^2) + ((16
*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*x*(a + c*x^2)^(3/2))/(64*c^2) + (11*d*e*(d
+ e*x)^2*(a + c*x^2)^(5/2))/(56*c) + (e*(d + e*x)^3*(a + c*x^2)^(5/2))/(8*c) + (
e*(4*d*(67*c*d^2 - 32*a*e^2) + 5*e*(26*c*d^2 - 7*a*e^2)*x)*(a + c*x^2)^(5/2))/(5
60*c^2) + (3*a^2*(16*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(128*c^(5/2))

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Rubi [A]  time = 0.651392, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{x \left (a+c x^2\right )^{3/2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{64 c^2}+\frac{3 a x \sqrt{a+c x^2} \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right )}{128 c^2}+\frac{3 a^2 \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (5 e x \left (26 c d^2-7 a e^2\right )+4 d \left (67 c d^2-32 a e^2\right )\right )}{560 c^2}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^3}{8 c}+\frac{11 d e \left (a+c x^2\right )^{5/2} (d+e x)^2}{56 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a + c*x^2)^(3/2),x]

[Out]

(3*a*(16*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*x*Sqrt[a + c*x^2])/(128*c^2) + ((16
*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*x*(a + c*x^2)^(3/2))/(64*c^2) + (11*d*e*(d
+ e*x)^2*(a + c*x^2)^(5/2))/(56*c) + (e*(d + e*x)^3*(a + c*x^2)^(5/2))/(8*c) + (
e*(4*d*(67*c*d^2 - 32*a*e^2) + 5*e*(26*c*d^2 - 7*a*e^2)*x)*(a + c*x^2)^(5/2))/(5
60*c^2) + (3*a^2*(16*c^2*d^4 - 16*a*c*d^2*e^2 + a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 57.4378, size = 246, normalized size = 0.96 \[ \frac{3 a^{2} \left (a^{2} e^{4} - 16 a c d^{2} e^{2} + 16 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} + \frac{3 a x \sqrt{a + c x^{2}} \left (a^{2} e^{4} - 16 a c d^{2} e^{2} + 16 c^{2} d^{4}\right )}{128 c^{2}} + \frac{11 d e \left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}{56 c} + \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{3}}{8 c} - \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}} \left (d \left (384 a e^{2} - 804 c d^{2}\right ) + 15 e x \left (7 a e^{2} - 26 c d^{2}\right )\right )}{1680 c^{2}} + \frac{x \left (a + c x^{2}\right )^{\frac{3}{2}} \left (a^{2} e^{4} - 16 a c d^{2} e^{2} + 16 c^{2} d^{4}\right )}{64 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(c*x**2+a)**(3/2),x)

[Out]

3*a**2*(a**2*e**4 - 16*a*c*d**2*e**2 + 16*c**2*d**4)*atanh(sqrt(c)*x/sqrt(a + c*
x**2))/(128*c**(5/2)) + 3*a*x*sqrt(a + c*x**2)*(a**2*e**4 - 16*a*c*d**2*e**2 + 1
6*c**2*d**4)/(128*c**2) + 11*d*e*(a + c*x**2)**(5/2)*(d + e*x)**2/(56*c) + e*(a
+ c*x**2)**(5/2)*(d + e*x)**3/(8*c) - e*(a + c*x**2)**(5/2)*(d*(384*a*e**2 - 804
*c*d**2) + 15*e*x*(7*a*e**2 - 26*c*d**2))/(1680*c**2) + x*(a + c*x**2)**(3/2)*(a
**2*e**4 - 16*a*c*d**2*e**2 + 16*c**2*d**4)/(64*c**2)

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Mathematica [A]  time = 0.296303, size = 231, normalized size = 0.91 \[ \frac{105 a^2 \left (a^2 e^4-16 a c d^2 e^2+16 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} \sqrt{a+c x^2} \left (-a^3 e^3 (1024 d+105 e x)+2 a^2 c e \left (1792 d^3+840 d^2 e x+256 d e^2 x^2+35 e^3 x^3\right )+8 a c^2 x \left (350 d^4+896 d^3 e x+980 d^2 e^2 x^2+512 d e^3 x^3+105 e^4 x^4\right )+16 c^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )}{4480 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(a + c*x^2)^(3/2),x]

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(-(a^3*e^3*(1024*d + 105*e*x)) + 2*a^2*c*e*(1792*d^3 +
840*d^2*e*x + 256*d*e^2*x^2 + 35*e^3*x^3) + 16*c^3*x^3*(70*d^4 + 224*d^3*e*x + 2
80*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 8*a*c^2*x*(350*d^4 + 896*d^3*e*x
+ 980*d^2*e^2*x^2 + 512*d*e^3*x^3 + 105*e^4*x^4)) + 105*a^2*(16*c^2*d^4 - 16*a*c
*d^2*e^2 + a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(4480*c^(5/2))

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Maple [A]  time = 0.018, size = 322, normalized size = 1.3 \[{\frac{{d}^{4}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{4}ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{d}^{4}{a}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{4}{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{4}ax}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{e}^{4}x}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}{a}^{3}x}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,{e}^{4}{a}^{4}}{128}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{8\,d{e}^{3}a}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}{e}^{2}x}{c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{2}{e}^{2}ax}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{2}x}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{3}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(c*x^2+a)^(3/2),x)

[Out]

1/4*d^4*x*(c*x^2+a)^(3/2)+3/8*d^4*a*x*(c*x^2+a)^(1/2)+3/8*d^4*a^2/c^(1/2)*ln(c^(
1/2)*x+(c*x^2+a)^(1/2))+1/8*e^4*x^3*(c*x^2+a)^(5/2)/c-1/16*e^4*a/c^2*x*(c*x^2+a)
^(5/2)+1/64*e^4*a^2/c^2*x*(c*x^2+a)^(3/2)+3/128*e^4*a^3/c^2*x*(c*x^2+a)^(1/2)+3/
128*e^4*a^4/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4/7*d*e^3*x^2*(c*x^2+a)^(5/2)/
c-8/35*d*e^3*a/c^2*(c*x^2+a)^(5/2)+d^2*e^2*x*(c*x^2+a)^(5/2)/c-1/4*d^2*e^2*a/c*x
*(c*x^2+a)^(3/2)-3/8*d^2*e^2*a^2/c*x*(c*x^2+a)^(1/2)-3/8*d^2*e^2*a^3/c^(3/2)*ln(
c^(1/2)*x+(c*x^2+a)^(1/2))+4/5*d^3*e*(c*x^2+a)^(5/2)/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((c*x^2 + a)^(3/2)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290216, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (560 \, c^{3} e^{4} x^{7} + 2560 \, c^{3} d e^{3} x^{6} + 3584 \, a^{2} c d^{3} e - 1024 \, a^{3} d e^{3} + 280 \,{\left (16 \, c^{3} d^{2} e^{2} + 3 \, a c^{2} e^{4}\right )} x^{5} + 512 \,{\left (7 \, c^{3} d^{3} e + 8 \, a c^{2} d e^{3}\right )} x^{4} + 70 \,{\left (16 \, c^{3} d^{4} + 112 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{3} + 512 \,{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} + 35 \,{\left (80 \, a c^{2} d^{4} + 48 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 105 \,{\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{8960 \, c^{\frac{5}{2}}}, \frac{{\left (560 \, c^{3} e^{4} x^{7} + 2560 \, c^{3} d e^{3} x^{6} + 3584 \, a^{2} c d^{3} e - 1024 \, a^{3} d e^{3} + 280 \,{\left (16 \, c^{3} d^{2} e^{2} + 3 \, a c^{2} e^{4}\right )} x^{5} + 512 \,{\left (7 \, c^{3} d^{3} e + 8 \, a c^{2} d e^{3}\right )} x^{4} + 70 \,{\left (16 \, c^{3} d^{4} + 112 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{3} + 512 \,{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} + 35 \,{\left (80 \, a c^{2} d^{4} + 48 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 105 \,{\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{4480 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(e*x + d)^4,x, algorithm="fricas")

[Out]

[1/8960*(2*(560*c^3*e^4*x^7 + 2560*c^3*d*e^3*x^6 + 3584*a^2*c*d^3*e - 1024*a^3*d
*e^3 + 280*(16*c^3*d^2*e^2 + 3*a*c^2*e^4)*x^5 + 512*(7*c^3*d^3*e + 8*a*c^2*d*e^3
)*x^4 + 70*(16*c^3*d^4 + 112*a*c^2*d^2*e^2 + a^2*c*e^4)*x^3 + 512*(14*a*c^2*d^3*
e + a^2*c*d*e^3)*x^2 + 35*(80*a*c^2*d^4 + 48*a^2*c*d^2*e^2 - 3*a^3*e^4)*x)*sqrt(
c*x^2 + a)*sqrt(c) + 105*(16*a^2*c^2*d^4 - 16*a^3*c*d^2*e^2 + a^4*e^4)*log(-2*sq
rt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/c^(5/2), 1/4480*((560*c^3*e^4*x^7 +
2560*c^3*d*e^3*x^6 + 3584*a^2*c*d^3*e - 1024*a^3*d*e^3 + 280*(16*c^3*d^2*e^2 + 3
*a*c^2*e^4)*x^5 + 512*(7*c^3*d^3*e + 8*a*c^2*d*e^3)*x^4 + 70*(16*c^3*d^4 + 112*a
*c^2*d^2*e^2 + a^2*c*e^4)*x^3 + 512*(14*a*c^2*d^3*e + a^2*c*d*e^3)*x^2 + 35*(80*
a*c^2*d^4 + 48*a^2*c*d^2*e^2 - 3*a^3*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) + 105*(16*
a^2*c^2*d^4 - 16*a^3*c*d^2*e^2 + a^4*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(s
qrt(-c)*c^2)]

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Sympy [A]  time = 90.6372, size = 734, normalized size = 2.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(c*x**2+a)**(3/2),x)

[Out]

-3*a**(7/2)*e**4*x/(128*c**2*sqrt(1 + c*x**2/a)) + 3*a**(5/2)*d**2*e**2*x/(8*c*s
qrt(1 + c*x**2/a)) - a**(5/2)*e**4*x**3/(128*c*sqrt(1 + c*x**2/a)) + a**(3/2)*d*
*4*x*sqrt(1 + c*x**2/a)/2 + a**(3/2)*d**4*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(3/2)
*d**2*e**2*x**3/(8*sqrt(1 + c*x**2/a)) + 13*a**(3/2)*e**4*x**5/(64*sqrt(1 + c*x*
*2/a)) + 3*sqrt(a)*c*d**4*x**3/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*d**2*e**2*x
**5/(4*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*c*e**4*x**7/(16*sqrt(1 + c*x**2/a)) + 3*a
**4*e**4*asinh(sqrt(c)*x/sqrt(a))/(128*c**(5/2)) - 3*a**3*d**2*e**2*asinh(sqrt(c
)*x/sqrt(a))/(8*c**(3/2)) + 3*a**2*d**4*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + 4
*a*d**3*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True
)) + 4*a*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)) + 4
*c*d**3*e*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)) + 4*c*d*e
**3*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)
/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)
), (sqrt(a)*x**6/6, True)) + c**2*d**4*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**
2*d**2*e**2*x**7/(sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*e**4*x**9/(8*sqrt(a)*sqrt(1
+ c*x**2/a))

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GIAC/XCAS [A]  time = 0.235302, size = 374, normalized size = 1.47 \[ \frac{1}{4480} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (2 \,{\left (7 \, c x e^{4} + 32 \, c d e^{3}\right )} x + \frac{7 \,{\left (16 \, c^{7} d^{2} e^{2} + 3 \, a c^{6} e^{4}\right )}}{c^{6}}\right )} x + \frac{64 \,{\left (7 \, c^{7} d^{3} e + 8 \, a c^{6} d e^{3}\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (16 \, c^{7} d^{4} + 112 \, a c^{6} d^{2} e^{2} + a^{2} c^{5} e^{4}\right )}}{c^{6}}\right )} x + \frac{256 \,{\left (14 \, a c^{6} d^{3} e + a^{2} c^{5} d e^{3}\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (80 \, a c^{6} d^{4} + 48 \, a^{2} c^{5} d^{2} e^{2} - 3 \, a^{3} c^{4} e^{4}\right )}}{c^{6}}\right )} x + \frac{512 \,{\left (7 \, a^{2} c^{5} d^{3} e - 2 \, a^{3} c^{4} d e^{3}\right )}}{c^{6}}\right )} - \frac{3 \,{\left (16 \, a^{2} c^{2} d^{4} - 16 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(e*x + d)^4,x, algorithm="giac")

[Out]

1/4480*sqrt(c*x^2 + a)*((2*((4*(5*(2*(7*c*x*e^4 + 32*c*d*e^3)*x + 7*(16*c^7*d^2*
e^2 + 3*a*c^6*e^4)/c^6)*x + 64*(7*c^7*d^3*e + 8*a*c^6*d*e^3)/c^6)*x + 35*(16*c^7
*d^4 + 112*a*c^6*d^2*e^2 + a^2*c^5*e^4)/c^6)*x + 256*(14*a*c^6*d^3*e + a^2*c^5*d
*e^3)/c^6)*x + 35*(80*a*c^6*d^4 + 48*a^2*c^5*d^2*e^2 - 3*a^3*c^4*e^4)/c^6)*x + 5
12*(7*a^2*c^5*d^3*e - 2*a^3*c^4*d*e^3)/c^6) - 3/128*(16*a^2*c^2*d^4 - 16*a^3*c*d
^2*e^2 + a^4*e^4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)