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

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

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Rubi [A]  time = 0.42, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {832, 779, 621, 206} \begin {gather*} \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} \end {gather*}

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)*ArcT
anh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(11/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^3 \left (-4 a B-\frac {1}{2} (9 b B-10 A c) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^2 \left (\frac {3}{2} a (9 b B-10 A c)+\frac {1}{4} \left (63 b^2 B-70 A b c-64 a B c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x \left (-\frac {1}{2} a \left (63 b^2 B-70 A b c-64 a B c\right )-\frac {1}{8} \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (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 \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (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\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (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 \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (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\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^5}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (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 \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (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\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 225, normalized size = 0.80 \begin {gather*} \frac {\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 )}{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+x (b+c x)}}\right )}{256 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))))/(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 + x*(b + c*x)])])/(256*c^(11/2))

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IntegrateAlgebraic [A]  time = 0.92, size = 244, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2+2200 a A b c^2-720 a A c^3 x-2940 a b^2 B c+1288 a b B c^2 x-512 a B c^3 x^2-1050 A b^3 c+700 A b^2 c^2 x-560 A b c^3 x^2+480 A c^4 x^3+945 b^4 B-630 b^3 B c x+504 b^2 B c^2 x^2-432 b B c^3 x^3+384 B c^4 x^4\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 ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{256 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(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 - 630*b^3*
B*c*x + 700*A*b^2*c^2*x + 1288*a*b*B*c^2*x - 720*a*A*c^3*x + 504*b^2*B*c^2*x^2 - 560*A*b*c^3*x^2 - 512*a*B*c^3
*x^2 - 432*b*B*c^3*x^3 + 480*A*c^4*x^3 + 384*B*c^4*x^4))/(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)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(256*c^(
11/2))

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fricas [A]  time = 0.52, size = 519, normalized size = 1.85 \begin {gather*} \left [-\frac {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 )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {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 )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \end {gather*}

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, algorithm="fricas")

[Out]

[-1/7680*(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)*sqrt(c)*log(-8
*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*B*c^5*x^4 + 945*B*b^4
*c + 8*(128*B*a^2 + 275*A*a*b)*c^3 - 48*(9*B*b*c^4 - 10*A*c^5)*x^3 - 210*(14*B*a*b^2 + 5*A*b^3)*c^2 + 8*(63*B*
b^2*c^3 - 2*(32*B*a + 35*A*b)*c^4)*x^2 - 2*(315*B*b^3*c^2 + 360*A*a*c^4 - 14*(46*B*a*b + 25*A*b^2)*c^3)*x)*sqr
t(c*x^2 + b*x + a))/c^6, 1/3840*(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)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(384*B*c^
5*x^4 + 945*B*b^4*c + 8*(128*B*a^2 + 275*A*a*b)*c^3 - 48*(9*B*b*c^4 - 10*A*c^5)*x^3 - 210*(14*B*a*b^2 + 5*A*b^
3)*c^2 + 8*(63*B*b^2*c^3 - 2*(32*B*a + 35*A*b)*c^4)*x^2 - 2*(315*B*b^3*c^2 + 360*A*a*c^4 - 14*(46*B*a*b + 25*A
*b^2)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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giac [A]  time = 0.26, size = 249, normalized size = 0.89 \begin {gather*} \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 )} \log \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}}} \end {gather*}

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, 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 - 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)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b
))/c^(11/2)

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maple [B]  time = 0.07, size = 531, normalized size = 1.89 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, B \,x^{4}}{5 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,x^{3}}{4 c}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, B b \,x^{3}}{40 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A b \,x^{2}}{24 c^{2}}-\frac {4 \sqrt {c \,x^{2}+b x +a}\, B a \,x^{2}}{15 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x^{2}}{80 c^{3}}+\frac {3 A \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {15 A a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {15 B \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 B a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {9}{2}}}-\frac {63 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a x}{8 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{96 c^{3}}+\frac {161 \sqrt {c \,x^{2}+b x +a}\, B a b x}{240 c^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} x}{64 c^{4}}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, A a b}{48 c^{3}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c^{4}}+\frac {8 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2}}{15 c^{3}}-\frac {49 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2}}{32 c^{4}}+\frac {63 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{128 c^{5}} \end {gather*}

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]

1/5*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)-2
1/64*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((c*x+1/2*b)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+35/32*B*b^3/c^(9/2)*a*ln((c*x+1/2*b)/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((c*x+1/2*b)/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)+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((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*A*b^2/c^(7/2)*a*ln((c*x+
1/2*b)/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((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (A + B x\right )}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

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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