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

Optimal. Leaf size=189 \[ \frac{5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]

[Out]

(5*a^2*(8*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*c*d^2 - a*e^2)*x*(
a + c*x^2)^(3/2))/(192*c) + ((8*c*d^2 - a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (9*
d*e*(a + c*x^2)^(7/2))/(56*c) + (e*(d + e*x)*(a + c*x^2)^(7/2))/(8*c) + (5*a^3*(
8*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

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Rubi [A]  time = 0.23335, antiderivative size = 189, 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 \[ \frac{5 a^3 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} \left (8 c d^2-a e^2\right )}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right )}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right )}{192 c}+\frac{9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(8*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*c*d^2 - a*e^2)*x*(
a + c*x^2)^(3/2))/(192*c) + ((8*c*d^2 - a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (9*
d*e*(a + c*x^2)^(7/2))/(56*c) + (e*(d + e*x)*(a + c*x^2)^(7/2))/(8*c) + (5*a^3*(
8*c*d^2 - 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 = 25.7264, size = 170, normalized size = 0.9 \[ - \frac{5 a^{3} \left (a e^{2} - 8 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{128 c^{\frac{3}{2}}} - \frac{5 a^{2} x \sqrt{a + c x^{2}} \left (a e^{2} - 8 c d^{2}\right )}{128 c} - \frac{5 a x \left (a + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} - 8 c d^{2}\right )}{192 c} + \frac{9 d e \left (a + c x^{2}\right )^{\frac{7}{2}}}{56 c} + \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (d + e x\right )}{8 c} - \frac{x \left (a + c x^{2}\right )^{\frac{5}{2}} \left (a e^{2} - 8 c d^{2}\right )}{48 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**3*(a*e**2 - 8*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(128*c**(3/2)) - 5
*a**2*x*sqrt(a + c*x**2)*(a*e**2 - 8*c*d**2)/(128*c) - 5*a*x*(a + c*x**2)**(3/2)
*(a*e**2 - 8*c*d**2)/(192*c) + 9*d*e*(a + c*x**2)**(7/2)/(56*c) + e*(a + c*x**2)
**(7/2)*(d + e*x)/(8*c) - x*(a + c*x**2)**(5/2)*(a*e**2 - 8*c*d**2)/(48*c)

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Mathematica [A]  time = 0.202597, size = 162, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (3 a^3 e (256 d+35 e x)+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )-105 a^3 \left (a e^2-8 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2688 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e
*x + 21*e^2*x^2) + 8*a*c^2*x^3*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(
924*d^2 + 1152*d*e*x + 413*e^2*x^2)) - 105*a^3*(-8*c*d^2 + a*e^2)*Log[c*x + Sqrt
[c]*Sqrt[a + c*x^2]])/(2688*c^(3/2))

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Maple [A]  time = 0.01, size = 200, normalized size = 1.1 \[{\frac{{d}^{2}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{d}^{2}x}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}{d}^{2}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{e}^{2}x}{48\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}{e}^{2}x}{192\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}{a}^{3}x}{128\,c}\sqrt{c{x}^{2}+a}}-{\frac{5\,{e}^{2}{a}^{4}}{128}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,de}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*d^2*x*(c*x^2+a)^(5/2)+5/24*d^2*a*x*(c*x^2+a)^(3/2)+5/16*d^2*a^2*x*(c*x^2+a)^
(1/2)+5/16*d^2*a^3/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/8*e^2*x*(c*x^2+a)^(7/
2)/c-1/48*e^2*a/c*x*(c*x^2+a)^(5/2)-5/192*e^2*a^2/c*x*(c*x^2+a)^(3/2)-5/128*e^2*
a^3/c*x*(c*x^2+a)^(1/2)-5/128*e^2*a^4/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+2/7*
d*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255357, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (336 \, c^{3} e^{2} x^{7} + 768 \, c^{3} d e x^{6} + 2304 \, a c^{2} d e x^{4} + 2304 \, a^{2} c d e x^{2} + 56 \,{\left (8 \, c^{3} d^{2} + 17 \, a c^{2} e^{2}\right )} x^{5} + 768 \, a^{3} d e + 14 \,{\left (104 \, a c^{2} d^{2} + 59 \, a^{2} c e^{2}\right )} x^{3} + 21 \,{\left (88 \, a^{2} c d^{2} + 5 \, a^{3} e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 105 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{5376 \, c^{\frac{3}{2}}}, \frac{{\left (336 \, c^{3} e^{2} x^{7} + 768 \, c^{3} d e x^{6} + 2304 \, a c^{2} d e x^{4} + 2304 \, a^{2} c d e x^{2} + 56 \,{\left (8 \, c^{3} d^{2} + 17 \, a c^{2} e^{2}\right )} x^{5} + 768 \, a^{3} d e + 14 \,{\left (104 \, a c^{2} d^{2} + 59 \, a^{2} c e^{2}\right )} x^{3} + 21 \,{\left (88 \, a^{2} c d^{2} + 5 \, a^{3} e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 105 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2688 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/5376*(2*(336*c^3*e^2*x^7 + 768*c^3*d*e*x^6 + 2304*a*c^2*d*e*x^4 + 2304*a^2*c*
d*e*x^2 + 56*(8*c^3*d^2 + 17*a*c^2*e^2)*x^5 + 768*a^3*d*e + 14*(104*a*c^2*d^2 +
59*a^2*c*e^2)*x^3 + 21*(88*a^2*c*d^2 + 5*a^3*e^2)*x)*sqrt(c*x^2 + a)*sqrt(c) + 1
05*(8*a^3*c*d^2 - a^4*e^2)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/
c^(3/2), 1/2688*((336*c^3*e^2*x^7 + 768*c^3*d*e*x^6 + 2304*a*c^2*d*e*x^4 + 2304*
a^2*c*d*e*x^2 + 56*(8*c^3*d^2 + 17*a*c^2*e^2)*x^5 + 768*a^3*d*e + 14*(104*a*c^2*
d^2 + 59*a^2*c*e^2)*x^3 + 21*(88*a^2*c*d^2 + 5*a^3*e^2)*x)*sqrt(c*x^2 + a)*sqrt(
-c) + 105*(8*a^3*c*d^2 - a^4*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*
c)]

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Sympy [A]  time = 89.5961, size = 539, normalized size = 2.85 \[ \frac{5 a^{\frac{7}{2}} e^{2} x}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{5}{2}} d^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} d^{2} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} e^{2} x^{3}}{384 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} c d^{2} x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} c e^{2} x^{5}}{192 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 \sqrt{a} c^{2} d^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{23 \sqrt{a} c^{2} e^{2} x^{7}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{5 a^{4} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{3}{2}}} + \frac{5 a^{3} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + 2 a^{2} d e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 4 a c d e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 c^{2} d e \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{3} d^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{3} e^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(7/2)*e**2*x/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**2*x*sqrt(1 + c*x**2/a
)/2 + 3*a**(5/2)*d**2*x/(16*sqrt(1 + c*x**2/a)) + 133*a**(5/2)*e**2*x**3/(384*sq
rt(1 + c*x**2/a)) + 35*a**(3/2)*c*d**2*x**3/(48*sqrt(1 + c*x**2/a)) + 127*a**(3/
2)*c*e**2*x**5/(192*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**2*x**5/(24*sqrt(1 +
c*x**2/a)) + 23*sqrt(a)*c**2*e**2*x**7/(48*sqrt(1 + c*x**2/a)) - 5*a**4*e**2*as
inh(sqrt(c)*x/sqrt(a))/(128*c**(3/2)) + 5*a**3*d**2*asinh(sqrt(c)*x/sqrt(a))/(16
*sqrt(c)) + 2*a**2*d*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2
)/(3*c), True)) + 4*a*c*d*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*c**2*d*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**3*d**2*x**7/(6*sqrt(a)*sqrt(1
+ c*x**2/a)) + c**3*e**2*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.221687, size = 257, normalized size = 1.36 \[ \frac{1}{2688} \,{\left (\frac{768 \, a^{3} d e}{c} +{\left (2 \,{\left (1152 \, a^{2} d e +{\left (4 \,{\left (288 \, a c d e +{\left (6 \,{\left (7 \, c^{2} x e^{2} + 16 \, c^{2} d e\right )} x + \frac{7 \,{\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac{7 \,{\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac{21 \,{\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt{c x^{2} + a} - \frac{5 \,{\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2688*(768*a^3*d*e/c + (2*(1152*a^2*d*e + (4*(288*a*c*d*e + (6*(7*c^2*x*e^2 + 1
6*c^2*d*e)*x + 7*(8*c^8*d^2 + 17*a*c^7*e^2)/c^6)*x)*x + 7*(104*a*c^7*d^2 + 59*a^
2*c^6*e^2)/c^6)*x)*x + 21*(88*a^2*c^6*d^2 + 5*a^3*c^5*e^2)/c^6)*x)*sqrt(c*x^2 +
a) - 5/128*(8*a^3*c*d^2 - a^4*e^2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)