3.535 \(\int (d+e x)^3 \left (a+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=216 \[ \frac{5 a^3 d \left (8 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 d x \sqrt{a+c x^2} \left (8 c d^2-3 a e^2\right )}{128 c}+\frac{e \left (a+c x^2\right )^{7/2} \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right )}{504 c^2}+\frac{d x \left (a+c x^2\right )^{5/2} \left (8 c d^2-3 a e^2\right )}{48 c}+\frac{5 a d x \left (a+c x^2\right )^{3/2} \left (8 c d^2-3 a e^2\right )}{192 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

[Out]

(5*a^2*d*(8*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*d*(8*c*d^2 - 3*a*
e^2)*x*(a + c*x^2)^(3/2))/(192*c) + (d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(5/2))/
(48*c) + (e*(d + e*x)^2*(a + c*x^2)^(7/2))/(9*c) + (e*(16*(10*c*d^2 - a*e^2) + 7
7*c*d*e*x)*(a + c*x^2)^(7/2))/(504*c^2) + (5*a^3*d*(8*c*d^2 - 3*a*e^2)*ArcTanh[(
Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

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Rubi [A]  time = 0.412135, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{5 a^3 d \left (8 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 d x \sqrt{a+c x^2} \left (8 c d^2-3 a e^2\right )}{128 c}+\frac{e \left (a+c x^2\right )^{7/2} \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right )}{504 c^2}+\frac{d x \left (a+c x^2\right )^{5/2} \left (8 c d^2-3 a e^2\right )}{48 c}+\frac{5 a d x \left (a+c x^2\right )^{3/2} \left (8 c d^2-3 a e^2\right )}{192 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a + c*x^2)^(5/2),x]

[Out]

(5*a^2*d*(8*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*d*(8*c*d^2 - 3*a*
e^2)*x*(a + c*x^2)^(3/2))/(192*c) + (d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(5/2))/
(48*c) + (e*(d + e*x)^2*(a + c*x^2)^(7/2))/(9*c) + (e*(16*(10*c*d^2 - a*e^2) + 7
7*c*d*e*x)*(a + c*x^2)^(7/2))/(504*c^2) + (5*a^3*d*(8*c*d^2 - 3*a*e^2)*ArcTanh[(
Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

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Rubi in Sympy [A]  time = 33.4873, size = 206, normalized size = 0.95 \[ - \frac{5 a^{3} d \left (3 a e^{2} - 8 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{128 c^{\frac{3}{2}}} - \frac{5 a^{2} d x \sqrt{a + c x^{2}} \left (3 a e^{2} - 8 c d^{2}\right )}{128 c} - \frac{5 a d x \left (a + c x^{2}\right )^{\frac{3}{2}} \left (3 a e^{2} - 8 c d^{2}\right )}{192 c} - \frac{d x \left (a + c x^{2}\right )^{\frac{5}{2}} \left (3 a e^{2} - 8 c d^{2}\right )}{48 c} + \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (d + e x\right )^{2}}{9 c} - \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (16 a e^{2} - 160 c d^{2} - 77 c d e x\right )}{504 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+a)**(5/2),x)

[Out]

-5*a**3*d*(3*a*e**2 - 8*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(128*c**(3/2))
- 5*a**2*d*x*sqrt(a + c*x**2)*(3*a*e**2 - 8*c*d**2)/(128*c) - 5*a*d*x*(a + c*x*
*2)**(3/2)*(3*a*e**2 - 8*c*d**2)/(192*c) - d*x*(a + c*x**2)**(5/2)*(3*a*e**2 - 8
*c*d**2)/(48*c) + e*(a + c*x**2)**(7/2)*(d + e*x)**2/(9*c) - e*(a + c*x**2)**(7/
2)*(16*a*e**2 - 160*c*d**2 - 77*c*d*e*x)/(504*c**2)

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Mathematica [A]  time = 0.321854, size = 216, normalized size = 1. \[ \frac{\sqrt{a+c x^2} \left (-256 a^4 e^3+a^3 c e \left (3456 d^2+945 d e x+128 e^2 x^2\right )+6 a^2 c^2 x \left (924 d^3+1728 d^2 e x+1239 d e^2 x^2+320 e^3 x^3\right )+8 a c^3 x^3 \left (546 d^3+1296 d^2 e x+1071 d e^2 x^2+304 e^3 x^3\right )+16 c^4 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )-315 a^3 \sqrt{c} d \left (3 a e^2-8 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{8064 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + c*x^2]*(-256*a^4*e^3 + a^3*c*e*(3456*d^2 + 945*d*e*x + 128*e^2*x^2) +
16*c^4*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 8*a*c^3*x^3*(54
6*d^3 + 1296*d^2*e*x + 1071*d*e^2*x^2 + 304*e^3*x^3) + 6*a^2*c^2*x*(924*d^3 + 17
28*d^2*e*x + 1239*d*e^2*x^2 + 320*e^3*x^3)) - 315*a^3*Sqrt[c]*d*(-8*c*d^2 + 3*a*
e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(8064*c^2)

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Maple [A]  time = 0.012, size = 245, normalized size = 1.1 \[{\frac{{d}^{3}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{3}ax}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{3}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}{d}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a{e}^{3}}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,d{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{ad{e}^{2}x}{16\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}d{e}^{2}x}{64\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,d{e}^{2}{a}^{3}x}{128\,c}\sqrt{c{x}^{2}+a}}-{\frac{15\,d{e}^{2}{a}^{4}}{128}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{d}^{2}e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+a)^(5/2),x)

[Out]

1/6*d^3*x*(c*x^2+a)^(5/2)+5/24*d^3*a*x*(c*x^2+a)^(3/2)+5/16*d^3*a^2*x*(c*x^2+a)^
(1/2)+5/16*d^3*a^3/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/9*e^3*x^2*(c*x^2+a)^(
7/2)/c-2/63*e^3*a/c^2*(c*x^2+a)^(7/2)+3/8*d*e^2*x*(c*x^2+a)^(7/2)/c-1/16*d*e^2*a
/c*x*(c*x^2+a)^(5/2)-5/64*d*e^2*a^2/c*x*(c*x^2+a)^(3/2)-15/128*d*e^2*a^3/c*x*(c*
x^2+a)^(1/2)-15/128*d*e^2*a^4/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+3/7*d^2*e*(c
*x^2+a)^(7/2)/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)^(5/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275592, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (896 \, c^{4} e^{3} x^{8} + 3024 \, c^{4} d e^{2} x^{7} + 3456 \, a^{3} c d^{2} e - 256 \, a^{4} e^{3} + 128 \,{\left (27 \, c^{4} d^{2} e + 19 \, a c^{3} e^{3}\right )} x^{6} + 168 \,{\left (8 \, c^{4} d^{3} + 51 \, a c^{3} d e^{2}\right )} x^{5} + 384 \,{\left (27 \, a c^{3} d^{2} e + 5 \, a^{2} c^{2} e^{3}\right )} x^{4} + 42 \,{\left (104 \, a c^{3} d^{3} + 177 \, a^{2} c^{2} d e^{2}\right )} x^{3} + 128 \,{\left (81 \, a^{2} c^{2} d^{2} e + a^{3} c e^{3}\right )} x^{2} + 63 \,{\left (88 \, a^{2} c^{2} d^{3} + 15 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 315 \,{\left (8 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{16128 \, c^{\frac{5}{2}}}, \frac{{\left (896 \, c^{4} e^{3} x^{8} + 3024 \, c^{4} d e^{2} x^{7} + 3456 \, a^{3} c d^{2} e - 256 \, a^{4} e^{3} + 128 \,{\left (27 \, c^{4} d^{2} e + 19 \, a c^{3} e^{3}\right )} x^{6} + 168 \,{\left (8 \, c^{4} d^{3} + 51 \, a c^{3} d e^{2}\right )} x^{5} + 384 \,{\left (27 \, a c^{3} d^{2} e + 5 \, a^{2} c^{2} e^{3}\right )} x^{4} + 42 \,{\left (104 \, a c^{3} d^{3} + 177 \, a^{2} c^{2} d e^{2}\right )} x^{3} + 128 \,{\left (81 \, a^{2} c^{2} d^{2} e + a^{3} c e^{3}\right )} x^{2} + 63 \,{\left (88 \, a^{2} c^{2} d^{3} + 15 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 315 \,{\left (8 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{8064 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/16128*(2*(896*c^4*e^3*x^8 + 3024*c^4*d*e^2*x^7 + 3456*a^3*c*d^2*e - 256*a^4*e
^3 + 128*(27*c^4*d^2*e + 19*a*c^3*e^3)*x^6 + 168*(8*c^4*d^3 + 51*a*c^3*d*e^2)*x^
5 + 384*(27*a*c^3*d^2*e + 5*a^2*c^2*e^3)*x^4 + 42*(104*a*c^3*d^3 + 177*a^2*c^2*d
*e^2)*x^3 + 128*(81*a^2*c^2*d^2*e + a^3*c*e^3)*x^2 + 63*(88*a^2*c^2*d^3 + 15*a^3
*c*d*e^2)*x)*sqrt(c*x^2 + a)*sqrt(c) + 315*(8*a^3*c^2*d^3 - 3*a^4*c*d*e^2)*log(-
2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/c^(5/2), 1/8064*((896*c^4*e^3*x^
8 + 3024*c^4*d*e^2*x^7 + 3456*a^3*c*d^2*e - 256*a^4*e^3 + 128*(27*c^4*d^2*e + 19
*a*c^3*e^3)*x^6 + 168*(8*c^4*d^3 + 51*a*c^3*d*e^2)*x^5 + 384*(27*a*c^3*d^2*e + 5
*a^2*c^2*e^3)*x^4 + 42*(104*a*c^3*d^3 + 177*a^2*c^2*d*e^2)*x^3 + 128*(81*a^2*c^2
*d^2*e + a^3*c*e^3)*x^2 + 63*(88*a^2*c^2*d^3 + 15*a^3*c*d*e^2)*x)*sqrt(c*x^2 + a
)*sqrt(-c) + 315*(8*a^3*c^2*d^3 - 3*a^4*c*d*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 +
a)))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 95.67, size = 843, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+a)**(5/2),x)

[Out]

15*a**(7/2)*d*e**2*x/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**3*x*sqrt(1 + c*x**
2/a)/2 + 3*a**(5/2)*d**3*x/(16*sqrt(1 + c*x**2/a)) + 133*a**(5/2)*d*e**2*x**3/(1
28*sqrt(1 + c*x**2/a)) + 35*a**(3/2)*c*d**3*x**3/(48*sqrt(1 + c*x**2/a)) + 127*a
**(3/2)*c*d*e**2*x**5/(64*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**3*x**5/(24*sq
rt(1 + c*x**2/a)) + 23*sqrt(a)*c**2*d*e**2*x**7/(16*sqrt(1 + c*x**2/a)) - 15*a**
4*d*e**2*asinh(sqrt(c)*x/sqrt(a))/(128*c**(3/2)) + 5*a**3*d**3*asinh(sqrt(c)*x/s
qrt(a))/(16*sqrt(c)) + 3*a**2*d**2*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a +
c*x**2)**(3/2)/(3*c), True)) + a**2*e**3*Piecewise((-2*a**2*sqrt(a + c*x**2)/(1
5*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)) + 6*a*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)) + 2*a*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)) + 3*c**2*d**2*e*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*e**3*Piecewise((-16*a**4*sqrt(a + c*x**2)/(315*c**4) + 8*a*
*3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*x**2)/(105*c**2) +
a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqrt(a)*x*
*8/8, True)) + c**3*d**3*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c**3*d*e**2*x**
9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.219758, size = 378, normalized size = 1.75 \[ \frac{1}{8064} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (2 \,{\left (7 \,{\left (8 \, c^{2} x e^{3} + 27 \, c^{2} d e^{2}\right )} x + \frac{8 \,{\left (27 \, c^{9} d^{2} e + 19 \, a c^{8} e^{3}\right )}}{c^{7}}\right )} x + \frac{21 \,{\left (8 \, c^{9} d^{3} + 51 \, a c^{8} d e^{2}\right )}}{c^{7}}\right )} x + \frac{48 \,{\left (27 \, a c^{8} d^{2} e + 5 \, a^{2} c^{7} e^{3}\right )}}{c^{7}}\right )} x + \frac{21 \,{\left (104 \, a c^{8} d^{3} + 177 \, a^{2} c^{7} d e^{2}\right )}}{c^{7}}\right )} x + \frac{64 \,{\left (81 \, a^{2} c^{7} d^{2} e + a^{3} c^{6} e^{3}\right )}}{c^{7}}\right )} x + \frac{63 \,{\left (88 \, a^{2} c^{7} d^{3} + 15 \, a^{3} c^{6} d e^{2}\right )}}{c^{7}}\right )} x + \frac{128 \,{\left (27 \, a^{3} c^{6} d^{2} e - 2 \, a^{4} c^{5} e^{3}\right )}}{c^{7}}\right )} - \frac{5 \,{\left (8 \, a^{3} c d^{3} - 3 \, a^{4} d e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/8064*sqrt(c*x^2 + a)*((2*((4*((2*(7*(8*c^2*x*e^3 + 27*c^2*d*e^2)*x + 8*(27*c^9
*d^2*e + 19*a*c^8*e^3)/c^7)*x + 21*(8*c^9*d^3 + 51*a*c^8*d*e^2)/c^7)*x + 48*(27*
a*c^8*d^2*e + 5*a^2*c^7*e^3)/c^7)*x + 21*(104*a*c^8*d^3 + 177*a^2*c^7*d*e^2)/c^7
)*x + 64*(81*a^2*c^7*d^2*e + a^3*c^6*e^3)/c^7)*x + 63*(88*a^2*c^7*d^3 + 15*a^3*c
^6*d*e^2)/c^7)*x + 128*(27*a^3*c^6*d^2*e - 2*a^4*c^5*e^3)/c^7) - 5/128*(8*a^3*c*
d^3 - 3*a^4*d*e^2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)