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

Optimal. Leaf size=367 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}+\frac{\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{13/2}}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac{x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

[Out]

-((33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32
*a^2*A*c^3)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^6) + ((33*b^2*B - 42*A*b*
c - 32*a*B*c)*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) - ((11*b*B - 14*A*c)*x^3*(a
+ b*x + c*x^2)^(3/2))/(84*c^2) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + ((1155
*b^4*B - 1470*A*b^3*c - 3276*a*b^2*B*c + 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(
231*b^3*B - 294*A*b^2*c - 444*a*b*B*c + 280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))
/(13440*c^5) + ((b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b
^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/(2048*c^(13/2))

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Rubi [A]  time = 1.1863, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}+\frac{\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{13/2}}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac{x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

-((33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32
*a^2*A*c^3)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^6) + ((33*b^2*B - 42*A*b*
c - 32*a*B*c)*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) - ((11*b*B - 14*A*c)*x^3*(a
+ b*x + c*x^2)^(3/2))/(84*c^2) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + ((1155
*b^4*B - 1470*A*b^3*c - 3276*a*b^2*B*c + 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(
231*b^3*B - 294*A*b^2*c - 444*a*b*B*c + 280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))
/(13440*c^5) + ((b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b
^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/(2048*c^(13/2))

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Rubi in Sympy [A]  time = 112.025, size = 394, normalized size = 1.07 \[ \frac{B x^{4} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{7 c} + \frac{x^{3} \left (14 A c - 11 B b\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{84 c^{2}} + \frac{x^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 42 A b c - 32 B a c + 33 B b^{2}\right )}{280 c^{3}} + \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{1029 A a b c^{2}}{2} - \frac{2205 A b^{3} c}{8} + 192 B a^{2} c^{2} - \frac{2457 B a b^{2} c}{4} + \frac{3465 B b^{4}}{16} - \frac{9 c x \left (280 A a c^{2} - 294 A b^{2} c - 444 B a b c + 231 B b^{3}\right )}{8}\right )}{2520 c^{5}} - \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (- 32 A a^{2} c^{3} + 112 A a b^{2} c^{2} - 42 A b^{4} c + 80 B a^{2} b c^{2} - 120 B a b^{3} c + 33 B b^{5}\right )}{1024 c^{6}} + \frac{\left (- 4 a c + b^{2}\right ) \left (- 32 A a^{2} c^{3} + 112 A a b^{2} c^{2} - 42 A b^{4} c + 80 B a^{2} b c^{2} - 120 B a b^{3} c + 33 B b^{5}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{13}{2}}} \]

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*(a + b*x + c*x**2)**(3/2)/(7*c) + x**3*(14*A*c - 11*B*b)*(a + b*x + c*x**
2)**(3/2)/(84*c**2) + x**2*(a + b*x + c*x**2)**(3/2)*(-42*A*b*c - 32*B*a*c + 33*
B*b**2)/(280*c**3) + (a + b*x + c*x**2)**(3/2)*(1029*A*a*b*c**2/2 - 2205*A*b**3*
c/8 + 192*B*a**2*c**2 - 2457*B*a*b**2*c/4 + 3465*B*b**4/16 - 9*c*x*(280*A*a*c**2
- 294*A*b**2*c - 444*B*a*b*c + 231*B*b**3)/8)/(2520*c**5) - (b + 2*c*x)*sqrt(a
+ b*x + c*x**2)*(-32*A*a**2*c**3 + 112*A*a*b**2*c**2 - 42*A*b**4*c + 80*B*a**2*b
*c**2 - 120*B*a*b**3*c + 33*B*b**5)/(1024*c**6) + (-4*a*c + b**2)*(-32*A*a**2*c*
*3 + 112*A*a*b**2*c**2 - 42*A*b**4*c + 80*B*a**2*b*c**2 - 120*B*a*b**3*c + 33*B*
b**5)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2048*c**(13/2))

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Mathematica [A]  time = 0.588145, size = 349, normalized size = 0.95 \[ \frac{105 \left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^2 c^2 \left (2163 a^2 B-2 a c x (392 A+243 B x)+2 c^2 x^3 (63 A+44 B x)\right )-32 b c^3 \left (a^2 (791 A+397 B x)-2 a c x^2 (119 A+79 B x)+8 c^2 x^4 (7 A+5 B x)\right )-64 c^3 \left (128 a^3 B-a^2 c x (105 A+64 B x)+2 a c^2 x^3 (35 A+24 B x)+40 c^3 x^5 (7 A+6 B x)\right )+84 b^4 c (c x (35 A+22 B x)-260 a B)+48 b^3 c^2 \left (14 a (35 A+18 B x)-c x^2 (49 A+33 B x)\right )-210 b^5 c (21 A+11 B x)+3465 b^6 B\right )}{215040 c^{13/2}} \]

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)]*(3465*b^6*B - 210*b^5*c*(21*A + 11*B*x) + 84*b
^4*c*(-260*a*B + c*x*(35*A + 22*B*x)) + 48*b^3*c^2*(14*a*(35*A + 18*B*x) - c*x^2
*(49*A + 33*B*x)) - 64*c^3*(128*a^3*B + 40*c^3*x^5*(7*A + 6*B*x) + 2*a*c^2*x^3*(
35*A + 24*B*x) - a^2*c*x*(105*A + 64*B*x)) + 16*b^2*c^2*(2163*a^2*B + 2*c^2*x^3*
(63*A + 44*B*x) - 2*a*c*x*(392*A + 243*B*x)) - 32*b*c^3*(8*c^2*x^4*(7*A + 5*B*x)
- 2*a*c*x^2*(119*A + 79*B*x) + a^2*(791*A + 397*B*x))) + 105*(b^2 - 4*a*c)*(33*
b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A
*c^3)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(215040*c^(13/2))

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Maple [B]  time = 0.022, size = 872, normalized size = 2.4 \[ \text{result too large to display} \]

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]

35/128*B*b^3/c^(9/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/35*B*a/c^
2*x^2*(c*x^2+b*x+a)^(3/2)-15/64*A*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))+49/240*A*b/c^3*a*(c*x^2+b*x+a)^(3/2)-1/8*A*a/c^2*x*(c*x^2+b*x+a)^(
3/2)+1/16*A*a^2/c^2*(c*x^2+b*x+a)^(1/2)*x+1/32*A*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b+1
/7*B*x^4*(c*x^2+b*x+a)^(3/2)/c-5/32*B*b/c^(7/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))-11/84*B*b/c^2*x^3*(c*x^2+b*x+a)^(3/2)+33/280*B*b^2/c^3*x^2*(c*x^
2+b*x+a)^(3/2)-33/320*B*b^3/c^4*x*(c*x^2+b*x+a)^(3/2)-33/512*B*b^5/c^5*(c*x^2+b*
x+a)^(1/2)*x+15/128*B*b^4/c^5*a*(c*x^2+b*x+a)^(1/2)-39/160*B*b^2/c^4*a*(c*x^2+b*
x+a)^(3/2)-5/64*B*b^2/c^4*a^2*(c*x^2+b*x+a)^(1/2)+15/64*B*b^3/c^4*a*(c*x^2+b*x+a
)^(1/2)*x-7/32*A*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*x-5/32*B*b/c^3*a^2*(c*x^2+b*x+a)^
(1/2)*x+111/560*B*b/c^3*a*x*(c*x^2+b*x+a)^(3/2)-3/20*A*b/c^2*x^2*(c*x^2+b*x+a)^(
3/2)+21/160*A*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)+21/256*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)
*x+35/256*A*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+8/105*B*a^
2/c^3*(c*x^2+b*x+a)^(3/2)+11/128*B*b^4/c^5*(c*x^2+b*x+a)^(3/2)-33/1024*B*b^6/c^6
*(c*x^2+b*x+a)^(1/2)+33/2048*B*b^7/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))+1/6*A*x^3*(c*x^2+b*x+a)^(3/2)/c-7/64*A*b^3/c^4*(c*x^2+b*x+a)^(3/2)+21/51
2*A*b^5/c^5*(c*x^2+b*x+a)^(1/2)-21/1024*A*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))+1/16*A*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)-63/512*B*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/64*A*b^3
/c^4*a*(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(sqrt(c*x^2 + b*x + a)*(B*x + A)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.390903, 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 + b*x + a)*(B*x + A)*x^4,x, algorithm="fricas")

[Out]

[1/430080*(4*(15360*B*c^6*x^6 - 3465*B*b^6 + 1280*(B*b*c^5 + 14*A*c^6)*x^5 - 128
*(11*B*b^2*c^4 - 2*(12*B*a + 7*A*b)*c^5)*x^4 + 32*(256*B*a^3 + 791*A*a^2*b)*c^3
+ 16*(99*B*b^3*c^3 + 280*A*a*c^5 - 2*(158*B*a*b + 63*A*b^2)*c^4)*x^3 - 336*(103*
B*a^2*b^2 + 70*A*a*b^3)*c^2 - 8*(231*B*b^4*c^2 + 8*(64*B*a^2 + 119*A*a*b)*c^4 -
6*(162*B*a*b^2 + 49*A*b^3)*c^3)*x^2 + 210*(104*B*a*b^4 + 21*A*b^5)*c + 2*(1155*B
*b^5*c - 3360*A*a^2*c^4 + 16*(397*B*a^2*b + 392*A*a*b^2)*c^3 - 42*(144*B*a*b^3 +
35*A*b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*(33*B*b^7 + 128*A*a^3*c^4
- 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B
*a*b^5 + A*b^6)*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^(13/2), 1/215040*(2*(15360*B*c^6*x^6 - 3465*B*
b^6 + 1280*(B*b*c^5 + 14*A*c^6)*x^5 - 128*(11*B*b^2*c^4 - 2*(12*B*a + 7*A*b)*c^5
)*x^4 + 32*(256*B*a^3 + 791*A*a^2*b)*c^3 + 16*(99*B*b^3*c^3 + 280*A*a*c^5 - 2*(1
58*B*a*b + 63*A*b^2)*c^4)*x^3 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^2 - 8*(231*B*
b^4*c^2 + 8*(64*B*a^2 + 119*A*a*b)*c^4 - 6*(162*B*a*b^2 + 49*A*b^3)*c^3)*x^2 + 2
10*(104*B*a*b^4 + 21*A*b^5)*c + 2*(1155*B*b^5*c - 3360*A*a^2*c^4 + 16*(397*B*a^2
*b + 392*A*a*b^2)*c^3 - 42*(144*B*a*b^3 + 35*A*b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(-c) + 105*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 +
280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B*a*b^5 + A*b^6)*c)*arctan(1/2*(2*c*x +
b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^6)]

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

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 = 0.289772, size = 559, normalized size = 1.52 \[ \frac{1}{107520} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B x + \frac{B b c^{5} + 14 \, A c^{6}}{c^{6}}\right )} x - \frac{11 \, B b^{2} c^{4} - 24 \, B a c^{5} - 14 \, A b c^{5}}{c^{6}}\right )} x + \frac{99 \, B b^{3} c^{3} - 316 \, B a b c^{4} - 126 \, A b^{2} c^{4} + 280 \, A a c^{5}}{c^{6}}\right )} x - \frac{231 \, B b^{4} c^{2} - 972 \, B a b^{2} c^{3} - 294 \, A b^{3} c^{3} + 512 \, B a^{2} c^{4} + 952 \, A a b c^{4}}{c^{6}}\right )} x + \frac{1155 \, B b^{5} c - 6048 \, B a b^{3} c^{2} - 1470 \, A b^{4} c^{2} + 6352 \, B a^{2} b c^{3} + 6272 \, A a b^{2} c^{3} - 3360 \, A a^{2} c^{4}}{c^{6}}\right )} x - \frac{3465 \, B b^{6} - 21840 \, B a b^{4} c - 4410 \, A b^{5} c + 34608 \, B a^{2} b^{2} c^{2} + 23520 \, A a b^{3} c^{2} - 8192 \, B a^{3} c^{3} - 25312 \, A a^{2} b c^{3}}{c^{6}}\right )} - \frac{{\left (33 \, B b^{7} - 252 \, B a b^{5} c - 42 \, A b^{6} c + 560 \, B a^{2} b^{3} c^{2} + 280 \, A a b^{4} c^{2} - 320 \, B a^{3} b c^{3} - 480 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*x + (B*b*c^5 + 14*A*c^6)/c^
6)*x - (11*B*b^2*c^4 - 24*B*a*c^5 - 14*A*b*c^5)/c^6)*x + (99*B*b^3*c^3 - 316*B*a
*b*c^4 - 126*A*b^2*c^4 + 280*A*a*c^5)/c^6)*x - (231*B*b^4*c^2 - 972*B*a*b^2*c^3
- 294*A*b^3*c^3 + 512*B*a^2*c^4 + 952*A*a*b*c^4)/c^6)*x + (1155*B*b^5*c - 6048*B
*a*b^3*c^2 - 1470*A*b^4*c^2 + 6352*B*a^2*b*c^3 + 6272*A*a*b^2*c^3 - 3360*A*a^2*c
^4)/c^6)*x - (3465*B*b^6 - 21840*B*a*b^4*c - 4410*A*b^5*c + 34608*B*a^2*b^2*c^2
+ 23520*A*a*b^3*c^2 - 8192*B*a^3*c^3 - 25312*A*a^2*b*c^3)/c^6) - 1/2048*(33*B*b^
7 - 252*B*a*b^5*c - 42*A*b^6*c + 560*B*a^2*b^3*c^2 + 280*A*a*b^4*c^2 - 320*B*a^3
*b*c^3 - 480*A*a^2*b^2*c^3 + 128*A*a^3*c^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c) - b))/c^(13/2)