3.2362 \(\int \frac{(d+e x)^4}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=296 \[ \frac{\left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{9/2}}+\frac{e \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{192 c^4}+\frac{7 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x)^3 \sqrt{a+b x+c x^2}}{4 c} \]

[Out]

(7*e*(2*c*d - b*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (e*(d + e*x)^3*
Sqrt[a + b*x + c*x^2])/(4*c) + (e*(608*c^3*d^3 - 105*b^3*e^3 + 20*b*c*e^2*(24*b*
d + 11*a*e) - 8*c^2*d*e*(101*b*d + 64*a*e) + 2*c*e*(104*c^2*d^2 + 35*b^2*e^2 - 4
*c*e*(26*b*d + 9*a*e))*x)*Sqrt[a + b*x + c*x^2])/(192*c^4) + ((128*c^4*d^4 + 35*
b^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c*e^3*(4*b*d + 3*a*e) + 48*c^2*
e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/(128*c^(9/2))

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Rubi [A]  time = 1.00278, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{9/2}}+\frac{e \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (9 a e+26 b d)+35 b^2 e^2+104 c^2 d^2\right )-8 c^2 d e (64 a e+101 b d)+20 b c e^2 (11 a e+24 b d)-105 b^3 e^3+608 c^3 d^3\right )}{192 c^4}+\frac{7 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x)^3 \sqrt{a+b x+c x^2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

(7*e*(2*c*d - b*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (e*(d + e*x)^3*
Sqrt[a + b*x + c*x^2])/(4*c) + (e*(608*c^3*d^3 - 105*b^3*e^3 + 20*b*c*e^2*(24*b*
d + 11*a*e) - 8*c^2*d*e*(101*b*d + 64*a*e) + 2*c*e*(104*c^2*d^2 + 35*b^2*e^2 - 4
*c*e*(26*b*d + 9*a*e))*x)*Sqrt[a + b*x + c*x^2])/(192*c^4) + ((128*c^4*d^4 + 35*
b^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c*e^3*(4*b*d + 3*a*e) + 48*c^2*
e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/(128*c^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.361794, size = 247, normalized size = 0.83 \[ \frac{3 \left (48 c^2 e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )-40 b^2 c e^3 (3 a e+4 b d)-128 c^3 d^2 e (3 a e+2 b d)+35 b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{c} e \sqrt{a+x (b+c x)} \left (-8 c^2 e \left (a e (64 d+9 e x)+b \left (108 d^2+40 d e x+7 e^2 x^2\right )\right )+10 b c e^2 (22 a e+48 b d+7 b e x)-105 b^3 e^3+16 c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )}{384 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(-105*b^3*e^3 + 10*b*c*e^2*(48*b*d + 22*a*e +
7*b*e*x) + 16*c^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) - 8*c^2*e*(a
*e*(64*d + 9*e*x) + b*(108*d^2 + 40*d*e*x + 7*e^2*x^2))) + 3*(128*c^4*d^4 + 35*b
^4*e^4 - 128*c^3*d^2*e*(2*b*d + 3*a*e) - 40*b^2*c*e^3*(4*b*d + 3*a*e) + 48*c^2*e
^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b +
c*x)]])/(384*c^(9/2))

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Maple [B]  time = 0.016, size = 627, normalized size = 2.1 \[{{d}^{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{4}{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,b{e}^{4}{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{2}{e}^{4}x}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,{e}^{4}{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{4}{e}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{15\,a{b}^{2}{e}^{4}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{55\,b{e}^{4}a}{48\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,a{e}^{4}x}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}{e}^{4}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,d{e}^{3}bx}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,d{e}^{3}{b}^{2}}{2\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{3}d{e}^{3}}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+3\,{\frac{abd{e}^{3}}{{c}^{5/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{8\,ad{e}^{3}}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+3\,{\frac{{d}^{2}{e}^{2}x\sqrt{c{x}^{2}+bx+a}}{c}}-{\frac{9\,{d}^{2}{e}^{2}b}{2\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{9\,{d}^{2}{e}^{2}{b}^{2}}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-3\,{\frac{a{d}^{2}{e}^{2}}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }+4\,{\frac{{d}^{3}e\sqrt{c{x}^{2}+bx+a}}{c}}-2\,{\frac{{d}^{3}eb}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1/4*e^4*x^3/c*(c*x^2+b*x
+a)^(1/2)-7/24*e^4*b/c^2*x^2*(c*x^2+b*x+a)^(1/2)+35/96*e^4*b^2/c^3*x*(c*x^2+b*x+
a)^(1/2)-35/64*e^4*b^3/c^4*(c*x^2+b*x+a)^(1/2)+35/128*e^4*b^4/c^(9/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*e^4*b^2/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))+55/48*e^4*b/c^3*a*(c*x^2+b*x+a)^(1/2)-3/8*e^4*a/c^2*x*(c*x
^2+b*x+a)^(1/2)+3/8*e^4*a^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+
4/3*d*e^3*x^2/c*(c*x^2+b*x+a)^(1/2)-5/3*d*e^3*b/c^2*x*(c*x^2+b*x+a)^(1/2)+5/2*d*
e^3*b^2/c^3*(c*x^2+b*x+a)^(1/2)-5/4*d*e^3*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))+3*d*e^3*b/c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)-8/3*d*e^3*a/c^2*(c*x^2+b*x+a)^(1/2)+3*d^2*e^2*x/c*(c*x^2+b*x+a)^(1/2)-9/2*d^2*
e^2*b/c^2*(c*x^2+b*x+a)^(1/2)+9/4*d^2*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))-3*d^2*e^2*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)+4*d^3*e/c*(c*x^2+b*x+a)^(1/2)-2*d^3*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+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((e*x + d)^4/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.450255, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, c^{3} e^{4} x^{3} + 768 \, c^{3} d^{3} e - 864 \, b c^{2} d^{2} e^{2} + 32 \,{\left (15 \, b^{2} c - 16 \, a c^{2}\right )} d e^{3} - 5 \,{\left (21 \, b^{3} - 44 \, a b c\right )} e^{4} + 8 \,{\left (32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} x^{2} + 2 \,{\left (288 \, c^{3} d^{2} e^{2} - 160 \, b c^{2} d e^{3} +{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} e^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \,{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} +{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (48 \, c^{3} e^{4} x^{3} + 768 \, c^{3} d^{3} e - 864 \, b c^{2} d^{2} e^{2} + 32 \,{\left (15 \, b^{2} c - 16 \, a c^{2}\right )} d e^{3} - 5 \,{\left (21 \, b^{3} - 44 \, a b c\right )} e^{4} + 8 \,{\left (32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} x^{2} + 2 \,{\left (288 \, c^{3} d^{2} e^{2} - 160 \, b c^{2} d e^{3} +{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} e^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 96 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e^{2} - 32 \,{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d e^{3} +{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} e^{4}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*c^3*e^4*x^3 + 768*c^3*d^3*e - 864*b*c^2*d^2*e^2 + 32*(15*b^2*c - 1
6*a*c^2)*d*e^3 - 5*(21*b^3 - 44*a*b*c)*e^4 + 8*(32*c^3*d*e^3 - 7*b*c^2*e^4)*x^2
+ 2*(288*c^3*d^2*e^2 - 160*b*c^2*d*e^3 + (35*b^2*c - 36*a*c^2)*e^4)*x)*sqrt(c*x^
2 + b*x + a)*sqrt(c) + 3*(128*c^4*d^4 - 256*b*c^3*d^3*e + 96*(3*b^2*c^2 - 4*a*c^
3)*d^2*e^2 - 32*(5*b^3*c - 12*a*b*c^2)*d*e^3 + (35*b^4 - 120*a*b^2*c + 48*a^2*c^
2)*e^4)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^
2 + 4*a*c)*sqrt(c)))/c^(9/2), 1/384*(2*(48*c^3*e^4*x^3 + 768*c^3*d^3*e - 864*b*c
^2*d^2*e^2 + 32*(15*b^2*c - 16*a*c^2)*d*e^3 - 5*(21*b^3 - 44*a*b*c)*e^4 + 8*(32*
c^3*d*e^3 - 7*b*c^2*e^4)*x^2 + 2*(288*c^3*d^2*e^2 - 160*b*c^2*d*e^3 + (35*b^2*c
- 36*a*c^2)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(128*c^4*d^4 - 256*b*c^3*
d^3*e + 96*(3*b^2*c^2 - 4*a*c^3)*d^2*e^2 - 32*(5*b^3*c - 12*a*b*c^2)*d*e^3 + (35
*b^4 - 120*a*b^2*c + 48*a^2*c^2)*e^4)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^
2 + b*x + a)*c)))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226597, size = 373, normalized size = 1.26 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \, x{\left (\frac{6 \, x e^{4}}{c} + \frac{32 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}}{c^{4}}\right )} + \frac{288 \, c^{3} d^{2} e^{2} - 160 \, b c^{2} d e^{3} + 35 \, b^{2} c e^{4} - 36 \, a c^{2} e^{4}}{c^{4}}\right )} x + \frac{768 \, c^{3} d^{3} e - 864 \, b c^{2} d^{2} e^{2} + 480 \, b^{2} c d e^{3} - 512 \, a c^{2} d e^{3} - 105 \, b^{3} e^{4} + 220 \, a b c e^{4}}{c^{4}}\right )} - \frac{{\left (128 \, c^{4} d^{4} - 256 \, b c^{3} d^{3} e + 288 \, b^{2} c^{2} d^{2} e^{2} - 384 \, a c^{3} d^{2} e^{2} - 160 \, b^{3} c d e^{3} + 384 \, a b c^{2} d e^{3} + 35 \, b^{4} e^{4} - 120 \, a b^{2} c e^{4} + 48 \, a^{2} c^{2} e^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*x*(6*x*e^4/c + (32*c^3*d*e^3 - 7*b*c^2*e^4)/c^
4) + (288*c^3*d^2*e^2 - 160*b*c^2*d*e^3 + 35*b^2*c*e^4 - 36*a*c^2*e^4)/c^4)*x +
(768*c^3*d^3*e - 864*b*c^2*d^2*e^2 + 480*b^2*c*d*e^3 - 512*a*c^2*d*e^3 - 105*b^3
*e^4 + 220*a*b*c*e^4)/c^4) - 1/128*(128*c^4*d^4 - 256*b*c^3*d^3*e + 288*b^2*c^2*
d^2*e^2 - 384*a*c^3*d^2*e^2 - 160*b^3*c*d*e^3 + 384*a*b*c^2*d*e^3 + 35*b^4*e^4 -
120*a*b^2*c*e^4 + 48*a^2*c^2*e^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*sqrt(c) - b))/c^(9/2)