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

Optimal. Leaf size=180 \[ \frac{3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac{d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac{3 a d x \sqrt{a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

[Out]

(3*a*d*(2*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(16*c) + (d*(2*c*d^2 - a*e^2)*x*(a +
c*x^2)^(3/2))/(8*c) + (e*(d + e*x)^2*(a + c*x^2)^(5/2))/(7*c) + (e*(4*(8*c*d^2
- a*e^2) + 15*c*d*e*x)*(a + c*x^2)^(5/2))/(70*c^2) + (3*a^2*d*(2*c*d^2 - a*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(3/2))

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Rubi [A]  time = 0.346579, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac{d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac{3 a d x \sqrt{a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*a*d*(2*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(16*c) + (d*(2*c*d^2 - a*e^2)*x*(a +
c*x^2)^(3/2))/(8*c) + (e*(d + e*x)^2*(a + c*x^2)^(5/2))/(7*c) + (e*(4*(8*c*d^2
- a*e^2) + 15*c*d*e*x)*(a + c*x^2)^(5/2))/(70*c^2) + (3*a^2*d*(2*c*d^2 - a*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(3/2))

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Rubi in Sympy [A]  time = 30.4255, size = 165, normalized size = 0.92 \[ - \frac{3 a^{2} d \left (a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{3}{2}}} - \frac{3 a d x \sqrt{a + c x^{2}} \left (a e^{2} - 2 c d^{2}\right )}{16 c} - \frac{d x \left (a + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} - 2 c d^{2}\right )}{8 c} + \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}{7 c} - \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}} \left (12 a e^{2} - 96 c d^{2} - 45 c d e x\right )}{210 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*d*(a*e**2 - 2*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(16*c**(3/2)) -
3*a*d*x*sqrt(a + c*x**2)*(a*e**2 - 2*c*d**2)/(16*c) - d*x*(a + c*x**2)**(3/2)*(a
*e**2 - 2*c*d**2)/(8*c) + e*(a + c*x**2)**(5/2)*(d + e*x)**2/(7*c) - e*(a + c*x*
*2)**(5/2)*(12*a*e**2 - 96*c*d**2 - 45*c*d*e*x)/(210*c**2)

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Mathematica [A]  time = 0.233027, size = 174, normalized size = 0.97 \[ \frac{\sqrt{a+c x^2} \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )-105 a^2 \sqrt{c} d \left (a e^2-2 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{560 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(-32*a^3*e^3 + a^2*c*e*(336*d^2 + 105*d*e*x + 16*e^2*x^2) + 4*c
^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 2*a*c^2*x*(175*d^3 +
336*d^2*e*x + 245*d*e^2*x^2 + 64*e^3*x^3)) - 105*a^2*Sqrt[c]*d*(-2*c*d^2 + a*e^2
)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(560*c^2)

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Maple [A]  time = 0.011, size = 205, normalized size = 1.1 \[{\frac{{d}^{3}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{3}ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}{d}^{3}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{3}a}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{d{e}^{2}x}{2\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ad{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{2}d{e}^{2}x}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,d{e}^{2}{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{d}^{2}e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256717, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \,{\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \,{\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \,{\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \,{\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 105 \,{\left (2 \, a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{1120 \, c^{\frac{5}{2}}}, \frac{{\left (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \,{\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \,{\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \,{\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \,{\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 105 \,{\left (2 \, a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{560 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1120*(2*(80*c^3*e^3*x^6 + 280*c^3*d*e^2*x^5 + 336*a^2*c*d^2*e - 32*a^3*e^3 +
16*(21*c^3*d^2*e + 8*a*c^2*e^3)*x^4 + 70*(2*c^3*d^3 + 7*a*c^2*d*e^2)*x^3 + 16*(4
2*a*c^2*d^2*e + a^2*c*e^3)*x^2 + 35*(10*a*c^2*d^3 + 3*a^2*c*d*e^2)*x)*sqrt(c*x^2
+ a)*sqrt(c) + 105*(2*a^2*c^2*d^3 - a^3*c*d*e^2)*log(-2*sqrt(c*x^2 + a)*c*x - (
2*c*x^2 + a)*sqrt(c)))/c^(5/2), 1/560*((80*c^3*e^3*x^6 + 280*c^3*d*e^2*x^5 + 336
*a^2*c*d^2*e - 32*a^3*e^3 + 16*(21*c^3*d^2*e + 8*a*c^2*e^3)*x^4 + 70*(2*c^3*d^3
+ 7*a*c^2*d*e^2)*x^3 + 16*(42*a*c^2*d^2*e + a^2*c*e^3)*x^2 + 35*(10*a*c^2*d^3 +
3*a^2*c*d*e^2)*x)*sqrt(c*x^2 + a)*sqrt(-c) + 105*(2*a^2*c^2*d^3 - a^3*c*d*e^2)*a
rctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 48.099, size = 551, normalized size = 3.06 \[ \frac{3 a^{\frac{5}{2}} d e^{2} x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{3}{2}} d^{3} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d^{3} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} d e^{2} x^{3}}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d^{3} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 \sqrt{a} c d e^{2} x^{5}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a^{3} d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{3 a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + 3 a d^{2} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + a e^{3} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 3 c d^{2} e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + c e^{3} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{2} d^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{2} d e^{2} x^{7}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**(5/2)*d*e**2*x/(16*c*sqrt(1 + c*x**2/a)) + a**(3/2)*d**3*x*sqrt(1 + c*x**2/
a)/2 + a**(3/2)*d**3*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*d*e**2*x**3/(16*sqrt
(1 + c*x**2/a)) + 3*sqrt(a)*c*d**3*x**3/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*d*
e**2*x**5/(8*sqrt(1 + c*x**2/a)) - 3*a**3*d*e**2*asinh(sqrt(c)*x/sqrt(a))/(16*c*
*(3/2)) + 3*a**2*d**3*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + 3*a*d**2*e*Piecewis
e((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*e**3*Piecew
ise((-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*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)) + c*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**3*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*d*e**2*x**7/(2*sqrt(a)*s
qrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.219072, size = 286, normalized size = 1.59 \[ \frac{1}{560} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (2 \, c x e^{3} + 7 \, c d e^{2}\right )} x + \frac{2 \,{\left (21 \, c^{6} d^{2} e + 8 \, a c^{5} e^{3}\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (2 \, c^{6} d^{3} + 7 \, a c^{5} d e^{2}\right )}}{c^{5}}\right )} x + \frac{8 \,{\left (42 \, a c^{5} d^{2} e + a^{2} c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (10 \, a c^{5} d^{3} + 3 \, a^{2} c^{4} d e^{2}\right )}}{c^{5}}\right )} x + \frac{16 \,{\left (21 \, a^{2} c^{4} d^{2} e - 2 \, a^{3} c^{3} e^{3}\right )}}{c^{5}}\right )} - \frac{3 \,{\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/560*sqrt(c*x^2 + a)*((2*((4*(5*(2*c*x*e^3 + 7*c*d*e^2)*x + 2*(21*c^6*d^2*e + 8
*a*c^5*e^3)/c^5)*x + 35*(2*c^6*d^3 + 7*a*c^5*d*e^2)/c^5)*x + 8*(42*a*c^5*d^2*e +
a^2*c^4*e^3)/c^5)*x + 35*(10*a*c^5*d^3 + 3*a^2*c^4*d*e^2)/c^5)*x + 16*(21*a^2*c
^4*d^2*e - 2*a^3*c^3*e^3)/c^5) - 3/16*(2*a^2*c*d^3 - a^3*d*e^2)*ln(abs(-sqrt(c)*
x + sqrt(c*x^2 + a)))/c^(3/2)