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

Optimal. Leaf size=255 \[ -\frac{d \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2} e^6}+\frac{\left (a+c x^2\right )^{3/2} \left (47 c d^2-8 a e^2\right )}{60 c^2 e^3}-\frac{d^4 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}+\frac{d \sqrt{a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^5}-\frac{13 d \left (a+c x^2\right )^{3/2} (d+e x)}{20 c e^3}+\frac{\left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c e^3} \]

[Out]

(d*(8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(8*c*e^5) + ((47*c*d^2 - 8
*a*e^2)*(a + c*x^2)^(3/2))/(60*c^2*e^3) - (13*d*(d + e*x)*(a + c*x^2)^(3/2))/(20
*c*e^3) + ((d + e*x)^2*(a + c*x^2)^(3/2))/(5*c*e^3) - (d*(8*c^2*d^4 + 4*a*c*d^2*
e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(3/2)*e^6) - (d^4*Sqrt
[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^
6

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Rubi [A]  time = 1.10185, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{d \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2} e^6}+\frac{\left (a+c x^2\right )^{3/2} \left (47 c d^2-8 a e^2\right )}{60 c^2 e^3}-\frac{d^4 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}+\frac{d \sqrt{a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{8 c e^5}-\frac{13 d \left (a+c x^2\right )^{3/2} (d+e x)}{20 c e^3}+\frac{\left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c e^3} \]

Antiderivative was successfully verified.

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

[Out]

(d*(8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(8*c*e^5) + ((47*c*d^2 - 8
*a*e^2)*(a + c*x^2)^(3/2))/(60*c^2*e^3) - (13*d*(d + e*x)*(a + c*x^2)^(3/2))/(20
*c*e^3) + ((d + e*x)^2*(a + c*x^2)^(3/2))/(5*c*e^3) - (d*(8*c^2*d^4 + 4*a*c*d^2*
e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(3/2)*e^6) - (d^4*Sqrt
[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^
6

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Rubi in Sympy [A]  time = 65.1515, size = 287, normalized size = 1.13 \[ \frac{a^{2} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{2}} - \frac{a d x \sqrt{a + c x^{2}}}{8 c e^{2}} - \frac{a \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c^{2} e} - \frac{a d^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c} e^{4}} - \frac{\sqrt{c} d^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{6}} + \frac{d^{4} \sqrt{a + c x^{2}}}{e^{5}} - \frac{d^{4} \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{6}} - \frac{d^{3} x \sqrt{a + c x^{2}}}{2 e^{4}} - \frac{d x^{3} \sqrt{a + c x^{2}}}{4 e^{2}} + \frac{d^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c e^{3}} + \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{5 c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(8*c**(3/2)*e**2) - a*d*x*sqrt(a + c*x*
*2)/(8*c*e**2) - a*(a + c*x**2)**(3/2)/(3*c**2*e) - a*d**3*atanh(sqrt(c)*x/sqrt(
a + c*x**2))/(2*sqrt(c)*e**4) - sqrt(c)*d**5*atanh(sqrt(c)*x/sqrt(a + c*x**2))/e
**6 + d**4*sqrt(a + c*x**2)/e**5 - d**4*sqrt(a*e**2 + c*d**2)*atanh((a*e - c*d*x
)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/e**6 - d**3*x*sqrt(a + c*x**2)/(2*e*
*4) - d*x**3*sqrt(a + c*x**2)/(4*e**2) + d**2*(a + c*x**2)**(3/2)/(3*c*e**3) + (
a + c*x**2)**(5/2)/(5*c**2*e)

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Mathematica [A]  time = 0.413962, size = 250, normalized size = 0.98 \[ \frac{-15 \sqrt{c} d \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+e \sqrt{a+c x^2} \left (-16 a^2 e^4+a c e^2 \left (40 d^2-15 d e x+8 e^2 x^2\right )+2 c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-120 c^2 d^4 \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+120 c^2 d^4 \sqrt{a e^2+c d^2} \log (d+e x)}{120 c^2 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2]*(-16*a^2*e^4 + a*c*e^2*(40*d^2 - 15*d*e*x + 8*e^2*x^2) + 2*c^
2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 120*c^2*
d^4*Sqrt[c*d^2 + a*e^2]*Log[d + e*x] - 15*Sqrt[c]*d*(8*c^2*d^4 + 4*a*c*d^2*e^2 -
a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - 120*c^2*d^4*Sqrt[c*d^2 + a*e^2]*L
og[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(120*c^2*e^6)

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Maple [B]  time = 0.034, size = 560, normalized size = 2.2 \[{\frac{{x}^{2}}{5\,ce} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,a}{15\,e{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{2}}{3\,{e}^{3}c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{4}}{{e}^{5}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}-{\frac{{d}^{5}}{{e}^{6}}\sqrt{c}\ln \left ({1 \left ( -{\frac{cd}{e}}+c \left ( x+{\frac{d}{e}} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) }-{\frac{{d}^{4}a}{{e}^{5}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{6}c}{{e}^{7}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{3}x}{2\,{e}^{4}}\sqrt{c{x}^{2}+a}}-{\frac{{d}^{3}a}{2\,{e}^{4}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{dx}{4\,c{e}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{adx}{8\,c{e}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{d{a}^{2}}{8\,{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/e*x^2*(c*x^2+a)^(3/2)/c-2/15/e*a/c^2*(c*x^2+a)^(3/2)+1/3*d^2/e^3*(c*x^2+a)^(
3/2)/c+d^4/e^5*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-d^5/e^6*c^(
1/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^
2)^(1/2))-d^4/e^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d
/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^
(1/2))/(x+d/e))*a-d^6/e^7/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*
d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^
2)/e^2)^(1/2))/(x+d/e))*c-1/2*d^3/e^4*x*(c*x^2+a)^(1/2)-1/2*d^3/e^4*a/c^(1/2)*ln
(x*c^(1/2)+(c*x^2+a)^(1/2))-1/4*d/e^2*x*(c*x^2+a)^(3/2)/c+1/8*d/e^2*a/c*x*(c*x^2
+a)^(1/2)+1/8*d/e^2*a^2/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

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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)*x^4/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.80388, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(120*sqrt(c*d^2 + a*e^2)*c^(5/2)*d^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e
^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2
+ a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 120*c^
2*d^4*e + 40*a*c*d^2*e^3 - 16*a^2*e^5 + 8*(5*c^2*d^2*e^3 + a*c*e^5)*x^2 - 15*(4*
c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) - 15*(8*c^3*d^5 + 4*a*c^2*d^
3*e^2 - a^2*c*d*e^4)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/(c^(5/
2)*e^6), 1/240*(240*sqrt(-c*d^2 - a*e^2)*c^(5/2)*d^4*arctan((c*d*x - a*e)/(sqrt(
-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + 2*(24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 120*c
^2*d^4*e + 40*a*c*d^2*e^3 - 16*a^2*e^5 + 8*(5*c^2*d^2*e^3 + a*c*e^5)*x^2 - 15*(4
*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) - 15*(8*c^3*d^5 + 4*a*c^2*d
^3*e^2 - a^2*c*d*e^4)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/(c^(5
/2)*e^6), 1/120*(60*sqrt(c*d^2 + a*e^2)*sqrt(-c)*c^2*d^4*log((2*a*c*d*e*x - a*c*
d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e
)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + (24*c^2*e^5*x^4 - 30*c^2*d*e^4*x
^3 + 120*c^2*d^4*e + 40*a*c*d^2*e^3 - 16*a^2*e^5 + 8*(5*c^2*d^2*e^3 + a*c*e^5)*x
^2 - 15*(4*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) - 15*(8*c^3*d^5
+ 4*a*c^2*d^3*e^2 - a^2*c*d*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c
^2*e^6), 1/120*(120*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*c^2*d^4*arctan((c*d*x - a*e)/(
sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + (24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 12
0*c^2*d^4*e + 40*a*c*d^2*e^3 - 16*a^2*e^5 + 8*(5*c^2*d^2*e^3 + a*c*e^5)*x^2 - 15
*(4*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) - 15*(8*c^3*d^5 + 4*a*c
^2*d^3*e^2 - a^2*c*d*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c^2*e^6)
]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \sqrt{a + c x^{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*sqrt(a + c*x**2)/(d + e*x), x)

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GIAC/XCAS [A]  time = 0.298813, size = 340, normalized size = 1.33 \[ \frac{2 \,{\left (c d^{6} + a d^{4} e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-6\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (4 \, x e^{\left (-1\right )} - 5 \, d e^{\left (-2\right )}\right )} x + \frac{4 \,{\left (5 \, c^{3} d^{2} e^{18} + a c^{2} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x - \frac{15 \,{\left (4 \, c^{3} d^{3} e^{17} + a c^{2} d e^{19}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x + \frac{8 \,{\left (15 \, c^{3} d^{4} e^{16} + 5 \, a c^{2} d^{2} e^{18} - 2 \, a^{2} c e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} + \frac{{\left (8 \, c^{\frac{5}{2}} d^{5} + 4 \, a c^{\frac{3}{2}} d^{3} e^{2} - a^{2} \sqrt{c} d e^{4}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d^6 + a*d^4*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt
(-c*d^2 - a*e^2))*e^(-6)/sqrt(-c*d^2 - a*e^2) + 1/120*sqrt(c*x^2 + a)*((2*(3*(4*
x*e^(-1) - 5*d*e^(-2))*x + 4*(5*c^3*d^2*e^18 + a*c^2*e^20)*e^(-21)/c^3)*x - 15*(
4*c^3*d^3*e^17 + a*c^2*d*e^19)*e^(-21)/c^3)*x + 8*(15*c^3*d^4*e^16 + 5*a*c^2*d^2
*e^18 - 2*a^2*c*e^20)*e^(-21)/c^3) + 1/8*(8*c^(5/2)*d^5 + 4*a*c^(3/2)*d^3*e^2 -
a^2*sqrt(c)*d*e^4)*e^(-6)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^2