3.950 \(\int \frac{x^4 (A+B x)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=281 \[ \frac{\sqrt{a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac{\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac{x^3 \sqrt{a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c} \]

[Out]

((63*b^2*B - 70*A*b*c - 64*a*B*c)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) - ((9*b*B
- 10*A*c)*x^3*Sqrt[a + b*x + c*x^2])/(40*c^2) + (B*x^4*Sqrt[a + b*x + c*x^2])/(
5*c) + ((945*b^4*B - 1050*A*b^3*c - 2940*a*b^2*B*c + 2200*a*A*b*c^2 + 1024*a^2*B
*c^2 - 2*c*(315*b^3*B - 350*A*b^2*c - 644*a*b*B*c + 360*a*A*c^2)*x)*Sqrt[a + b*x
+ c*x^2])/(1920*c^5) - ((63*b^5*B - 70*A*b^4*c - 280*a*b^3*B*c + 240*a*A*b^2*c^
2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x
+ c*x^2])])/(256*c^(11/2))

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Rubi [A]  time = 0.966816, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac{\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac{x^3 \sqrt{a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

((63*b^2*B - 70*A*b*c - 64*a*B*c)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) - ((9*b*B
- 10*A*c)*x^3*Sqrt[a + b*x + c*x^2])/(40*c^2) + (B*x^4*Sqrt[a + b*x + c*x^2])/(
5*c) + ((945*b^4*B - 1050*A*b^3*c - 2940*a*b^2*B*c + 2200*a*A*b*c^2 + 1024*a^2*B
*c^2 - 2*c*(315*b^3*B - 350*A*b^2*c - 644*a*b*B*c + 360*a*A*c^2)*x)*Sqrt[a + b*x
+ c*x^2])/(1920*c^5) - ((63*b^5*B - 70*A*b^4*c - 280*a*b^3*B*c + 240*a*A*b^2*c^
2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x
+ c*x^2])])/(256*c^(11/2))

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Rubi in Sympy [A]  time = 89.9303, size = 299, normalized size = 1.06 \[ \frac{B x^{4} \sqrt{a + b x + c x^{2}}}{5 c} + \frac{x^{3} \left (10 A c - 9 B b\right ) \sqrt{a + b x + c x^{2}}}{40 c^{2}} + \frac{x^{2} \sqrt{a + b x + c x^{2}} \left (- 70 A b c - 64 B a c + 63 B b^{2}\right )}{240 c^{3}} + \frac{\sqrt{a + b x + c x^{2}} \left (\frac{275 A a b c^{2}}{2} - \frac{525 A b^{3} c}{8} + 64 B a^{2} c^{2} - \frac{735 B a b^{2} c}{4} + \frac{945 B b^{4}}{16} - \frac{c x \left (360 A a c^{2} - 350 A b^{2} c - 644 B a b c + 315 B b^{3}\right )}{8}\right )}{120 c^{5}} - \frac{\left (- 96 A a^{2} c^{3} + 240 A a b^{2} c^{2} - 70 A b^{4} c + 240 B a^{2} b c^{2} - 280 B a b^{3} c + 63 B b^{5}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**4*sqrt(a + b*x + c*x**2)/(5*c) + x**3*(10*A*c - 9*B*b)*sqrt(a + b*x + c*x**
2)/(40*c**2) + x**2*sqrt(a + b*x + c*x**2)*(-70*A*b*c - 64*B*a*c + 63*B*b**2)/(2
40*c**3) + sqrt(a + b*x + c*x**2)*(275*A*a*b*c**2/2 - 525*A*b**3*c/8 + 64*B*a**2
*c**2 - 735*B*a*b**2*c/4 + 945*B*b**4/16 - c*x*(360*A*a*c**2 - 350*A*b**2*c - 64
4*B*a*b*c + 315*B*b**3)/8)/(120*c**5) - (-96*A*a**2*c**3 + 240*A*a*b**2*c**2 - 7
0*A*b**4*c + 240*B*a**2*b*c**2 - 280*B*a*b**3*c + 63*B*b**5)*atanh((b + 2*c*x)/(
2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(256*c**(11/2))

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Mathematica [A]  time = 0.354202, size = 225, normalized size = 0.8 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 c^2 \left (64 a^2 B-a c x (45 A+32 B x)+6 c^2 x^3 (5 A+4 B x)\right )+28 b^2 c (c x (25 A+18 B x)-105 a B)+8 b c^2 \left (a (275 A+161 B x)-2 c x^2 (35 A+27 B x)\right )-210 b^3 c (5 A+3 B x)+945 b^4 B\right )-15 \left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3840 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4*B - 210*b^3*c*(5*A + 3*B*x) + 28*b^2*c
*(-105*a*B + c*x*(25*A + 18*B*x)) + 16*c^2*(64*a^2*B + 6*c^2*x^3*(5*A + 4*B*x) -
a*c*x*(45*A + 32*B*x)) + 8*b*c^2*(-2*c*x^2*(35*A + 27*B*x) + a*(275*A + 161*B*x
))) - 15*(63*b^5*B - 70*A*b^4*c - 280*a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*
c^2 - 96*a^2*A*c^3)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3840*c^(1
1/2))

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Maple [B]  time = 0.017, size = 531, normalized size = 1.9 \[{\frac{A{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,Ab{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,Ax{b}^{2}}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,A{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,A{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{15\,a{b}^{2}A}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{55\,abA}{48\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,aAx}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{x}^{4}B}{5\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{9\,Bb{x}^{3}}{40\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{21\,{b}^{2}B{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{21\,Bx{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{63\,{b}^{4}B}{128\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{63\,B{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{11}{2}}}}+{\frac{35\,Ba{b}^{3}}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{49\,a{b}^{2}B}{32\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{161\,Bxab}{240\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{15\,B{a}^{2}b}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{4\,aB{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{8\,{a}^{2}B}{15\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*A*x^3/c*(c*x^2+b*x+a)^(1/2)-7/24*A*b/c^2*x^2*(c*x^2+b*x+a)^(1/2)+35/96*A*b^2
/c^3*x*(c*x^2+b*x+a)^(1/2)-35/64*A*b^3/c^4*(c*x^2+b*x+a)^(1/2)+35/128*A*b^4/c^(9
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*A*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*A*b/c^3*a*(c*x^2+b*x+a)^(1/2)-3/8*A*a/c^
2*x*(c*x^2+b*x+a)^(1/2)+3/8*A*a^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))+1/5*B*x^4*(c*x^2+b*x+a)^(1/2)/c-9/40*B*b/c^2*x^3*(c*x^2+b*x+a)^(1/2)+21/80
*B*b^2/c^3*x^2*(c*x^2+b*x+a)^(1/2)-21/64*B*b^3/c^4*x*(c*x^2+b*x+a)^(1/2)+63/128*
B*b^4/c^5*(c*x^2+b*x+a)^(1/2)-63/256*B*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))+35/32*B*b^3/c^(9/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)-49/32*B*b^2/c^4*a*(c*x^2+b*x+a)^(1/2)+161/240*B*b/c^3*a*x*(c*x^2+b*x+a)^(1/2)-
15/16*B*b/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/15*B*a/c^2*x
^2*(c*x^2+b*x+a)^(1/2)+8/15*B*a^2/c^3*(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((B*x + A)*x^4/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.47446, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, B c^{4} x^{4} + 945 \, B b^{4} - 48 \,{\left (9 \, B b c^{3} - 10 \, A c^{4}\right )} x^{3} + 8 \,{\left (128 \, B a^{2} + 275 \, A a b\right )} c^{2} + 8 \,{\left (63 \, B b^{2} c^{2} - 2 \,{\left (32 \, B a + 35 \, A b\right )} c^{3}\right )} x^{2} - 210 \,{\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c - 2 \,{\left (315 \, B b^{3} c + 360 \, A a c^{3} - 14 \,{\left (46 \, B a b + 25 \, A b^{2}\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \,{\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{7680 \, c^{\frac{11}{2}}}, \frac{2 \,{\left (384 \, B c^{4} x^{4} + 945 \, B b^{4} - 48 \,{\left (9 \, B b c^{3} - 10 \, A c^{4}\right )} x^{3} + 8 \,{\left (128 \, B a^{2} + 275 \, A a b\right )} c^{2} + 8 \,{\left (63 \, B b^{2} c^{2} - 2 \,{\left (32 \, B a + 35 \, A b\right )} c^{3}\right )} x^{2} - 210 \,{\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c - 2 \,{\left (315 \, B b^{3} c + 360 \, A a c^{3} - 14 \,{\left (46 \, B a b + 25 \, A b^{2}\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \,{\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{3840 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*B*c^4*x^4 + 945*B*b^4 - 48*(9*B*b*c^3 - 10*A*c^4)*x^3 + 8*(128*B
*a^2 + 275*A*a*b)*c^2 + 8*(63*B*b^2*c^2 - 2*(32*B*a + 35*A*b)*c^3)*x^2 - 210*(14
*B*a*b^2 + 5*A*b^3)*c - 2*(315*B*b^3*c + 360*A*a*c^3 - 14*(46*B*a*b + 25*A*b^2)*
c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 15*(63*B*b^5 - 96*A*a^2*c^3 + 240*(B*a^2
*b + A*a*b^2)*c^2 - 70*(4*B*a*b^3 + A*b^4)*c)*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^(11/2), 1/3840*(2*(
384*B*c^4*x^4 + 945*B*b^4 - 48*(9*B*b*c^3 - 10*A*c^4)*x^3 + 8*(128*B*a^2 + 275*A
*a*b)*c^2 + 8*(63*B*b^2*c^2 - 2*(32*B*a + 35*A*b)*c^3)*x^2 - 210*(14*B*a*b^2 + 5
*A*b^3)*c - 2*(315*B*b^3*c + 360*A*a*c^3 - 14*(46*B*a*b + 25*A*b^2)*c^2)*x)*sqrt
(c*x^2 + b*x + a)*sqrt(-c) - 15*(63*B*b^5 - 96*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^
2)*c^2 - 70*(4*B*a*b^3 + A*b^4)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 +
b*x + a)*c)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(A + B*x)/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.295565, size = 336, normalized size = 1.2 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, B x}{c} - \frac{9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac{63 \, B b^{2} c^{2} - 64 \, B a c^{3} - 70 \, A b c^{3}}{c^{5}}\right )} x - \frac{315 \, B b^{3} c - 644 \, B a b c^{2} - 350 \, A b^{2} c^{2} + 360 \, A a c^{3}}{c^{5}}\right )} x + \frac{945 \, B b^{4} - 2940 \, B a b^{2} c - 1050 \, A b^{3} c + 1024 \, B a^{2} c^{2} + 2200 \, A a b c^{2}}{c^{5}}\right )} + \frac{{\left (63 \, B b^{5} - 280 \, B a b^{3} c - 70 \, A b^{4} c + 240 \, B a^{2} b c^{2} + 240 \, A a b^{2} c^{2} - 96 \, A a^{2} c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*B*x/c - (9*B*b*c^3 - 10*A*c^4)/c^5)*x +
(63*B*b^2*c^2 - 64*B*a*c^3 - 70*A*b*c^3)/c^5)*x - (315*B*b^3*c - 644*B*a*b*c^2
- 350*A*b^2*c^2 + 360*A*a*c^3)/c^5)*x + (945*B*b^4 - 2940*B*a*b^2*c - 1050*A*b^3
*c + 1024*B*a^2*c^2 + 2200*A*a*b*c^2)/c^5) + 1/256*(63*B*b^5 - 280*B*a*b^3*c - 7
0*A*b^4*c + 240*B*a^2*b*c^2 + 240*A*a*b^2*c^2 - 96*A*a^2*c^3)*ln(abs(-2*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)