3.9.56 \(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=217 \[ \frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{64 a^2 x^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

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Rubi [A]  time = 0.25, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {810, 843, 621, 206, 724} \begin {gather*} -\frac {\sqrt {a+b x+c x^2} \left (x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )+2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )\right )}{64 a^2 x^2}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

-((2*a*(8*a*b*B - 3*A*(b^2 - 4*a*c)) + (8*a*B*(b^2 + 8*a*c) - 3*A*(b^3 - 4*a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(
64*a^2*x^2) - ((6*a*A + (3*A*b + 8*a*B)*x)*(a + b*x + c*x^2)^(3/2))/(24*a*x^4) + ((8*a*b*B*(b^2 - 12*a*c) - 3*
A*(b^2 - 4*a*c)^2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(5/2)) + B*c^(3/2)*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^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 724

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

Rule 810

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

Rule 843

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

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}-\frac {\int \frac {\left (\frac {1}{2} \left (-8 a b B+3 A \left (b^2-4 a c\right )\right )-8 a B c x\right ) \sqrt {a+b x+c x^2}}{x^3} \, dx}{8 a}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\int \frac {\frac {1}{4} \left (-8 a b B \left (b^2-12 a c\right )+3 A \left (b^2-4 a c\right )^2\right )+32 a^2 B c^2 x}{x \sqrt {a+b x+c x^2}} \, dx}{32 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (B c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (2 B c^2\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 )+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{64 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 202, normalized size = 0.93 \begin {gather*} -\frac {\left (3 A \left (b^2-4 a c\right )^2+8 a b B \left (12 a c-b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{128 a^{5/2}}-\frac {\sqrt {a+x (b+c x)} \left (16 a^3 (3 A+4 B x)+8 a^2 x (3 A (3 b+5 c x)+2 B x (7 b+16 c x))+6 a b x^2 (A (b+10 c x)+4 b B x)-9 A b^3 x^3\right )}{192 a^2 x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

-1/192*(Sqrt[a + x*(b + c*x)]*(-9*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 6*a*b*x^2*(4*b*B*x + A*(b + 10*c*x)) + 8*
a^2*x*(3*A*(3*b + 5*c*x) + 2*B*x*(7*b + 16*c*x))))/(a^2*x^4) - ((3*A*(b^2 - 4*a*c)^2 + 8*a*b*B*(-b^2 + 12*a*c)
)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(128*a^(5/2)) + B*c^(3/2)*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + x*(b + c*x)])]

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IntegrateAlgebraic [A]  time = 1.73, size = 229, normalized size = 1.06 \begin {gather*} \frac {\left (-48 a^2 A c^2-96 a^2 b B c+24 a A b^2 c+8 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{64 a^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (-48 a^3 A-64 a^3 B x-72 a^2 A b x-120 a^2 A c x^2-112 a^2 b B x^2-256 a^2 B c x^3-6 a A b^2 x^2-60 a A b c x^3-24 a b^2 B x^3+9 A b^3 x^3\right )}{192 a^2 x^4}-B c^{3/2} \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-48*a^3*A - 72*a^2*A*b*x - 64*a^3*B*x - 6*a*A*b^2*x^2 - 112*a^2*b*B*x^2 - 120*a^2*A*c*
x^2 + 9*A*b^3*x^3 - 24*a*b^2*B*x^3 - 60*a*A*b*c*x^3 - 256*a^2*B*c*x^3))/(192*a^2*x^4) + ((-3*A*b^4 + 8*a*b^3*B
+ 24*a*A*b^2*c - 96*a^2*b*B*c - 48*a^2*A*c^2)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(64*a^
(5/2)) - B*c^(3/2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]]

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fricas [A]  time = 2.57, size = 1083, normalized size = 4.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(384*B*a^3*c^(3/2)*x^4*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) - 3*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt(a)*x^4*log(-(8*a*b*x + (b^2 +
4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*
b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*
b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), -1/768*(768*B*a^3*sqrt(-c)*c*x^4*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)) + 3*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c
)*sqrt(a)*x^4*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) +
4*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*
A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), 1/384*(192*B*a^3*c^(3/2)*x^4*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) - 3*(8*B*a*b^3 - 3*A*b^4 - 48
*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt(-a)*x^4*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a
*c*x^2 + a*b*x + a^2)) - 2*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*
a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), -1/384*(3
84*B*a^3*sqrt(-c)*c*x^4*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)) + 3*(8*
B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*sqrt(-a)*x^4*arctan(1/2*sqrt(c*x^2 + b*x + a)*(
b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) + 2*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^
2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)
)/(a^3*x^4)]

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giac [B]  time = 0.59, size = 1019, normalized size = 4.70 \begin {gather*} -B c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right ) - \frac {{\left (8 \, B a b^{3} - 3 \, A b^{4} - 96 \, B a^{2} b c + 24 \, A a b^{2} c - 48 \, A a^{2} c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{2}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} + 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b c + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{2} c + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} c^{2} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{2} b^{2} \sqrt {c} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} c^{\frac {3}{2}} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{2} b c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} c^{2} - 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{3} b^{2} \sqrt {c} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} b^{3} \sqrt {c} - 1536 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} + 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 1280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{4} b c^{\frac {3}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} b^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b^{4} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{5} b c + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{2} c + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} c^{2} - 512 \, B a^{6} c^{\frac {3}{2}}}{192 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{4} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x, algorithm="giac")

[Out]

-B*c^(3/2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c - b*sqrt(c))) - 1/64*(8*B*a*b^3 - 3*A*b^4 - 96*B*a
^2*b*c + 24*A*a*b^2*c - 48*A*a^2*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1
/192*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*b^4 + 480*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b*c + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c + 240*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^2*b^2*sqrt(c) + 76
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b*c^(3
/2) + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 -
480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c +
144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^3*b^2*sqrt
(c) + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^2*b^3*sqrt(c) - 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4
*B*a^4*c^(3/2) - 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^3*b^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3
*A*a^2*b^4 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A
*a^3*b^2*c + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*c^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
B*a^5*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^4*b*c^(3/2) + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a^4*b^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a
^5*b*c + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^2*c + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*c^
2 - 512*B*a^6*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^4*a^2)

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maple [B]  time = 0.08, size = 838, normalized size = 3.86 \begin {gather*} -\frac {3 A \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+\frac {3 A \,b^{2} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}-\frac {3 A \,b^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {5}{2}}}-\frac {3 B b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 \sqrt {a}}+\frac {B \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}+B \,c^{\frac {3}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b \,c^{2} x}{16 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} c x}{64 a^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,c^{2} x}{a}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{2} c x}{8 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,c^{2}}{8 a}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c}{32 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{64 a^{3}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b \,c^{2} x}{16 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3} c x}{64 a^{4}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b c}{4 a}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{8 a^{2}}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,c^{2} x}{3 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} c x}{24 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,c^{2}}{8 a^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} c}{32 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{4}}{64 a^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b c}{12 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3}}{24 a^{3}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b c}{16 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{3}}{64 a^{4} x}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B c}{3 a^{2} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{24 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2}}{32 a^{3} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{12 a^{2} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{8 a^{2} x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{4 a \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x)

[Out]

3/64*A/a^3*b^3*c*(c*x^2+b*x+a)^(1/2)*x+1/64*A/a^4*b^3*c*(c*x^2+b*x+a)^(3/2)*x-1/24*B/a^3*b^2*c*(c*x^2+b*x+a)^(
3/2)*x-1/8*B/a^2*b^2*c*(c*x^2+b*x+a)^(1/2)*x-3/16*A/a^3*b*c^2*(c*x^2+b*x+a)^(3/2)*x-3/16*A/a^2*b*c^2*(c*x^2+b*
x+a)^(1/2)*x+3/16*A/a^3*b*c/x*(c*x^2+b*x+a)^(5/2)+5/4*B/a*b*c*(c*x^2+b*x+a)^(1/2)-2/3*B*c/a^2/x*(c*x^2+b*x+a)^
(5/2)-1/3*B/a/x^3*(c*x^2+b*x+a)^(5/2)-1/24*B/a^3*b^3*(c*x^2+b*x+a)^(3/2)-1/8*B/a^2*b^3*(c*x^2+b*x+a)^(1/2)+1/1
6*B/a^(3/2)*b^3*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-3/128*A/a^(5/2)*b^4*ln((b*x+2*a+2*(c*x^2+b*x+a)^
(1/2)*a^(1/2))/x)+1/8*A*c^2/a^2*(c*x^2+b*x+a)^(3/2)+3/8*A*c^2/a*(c*x^2+b*x+a)^(1/2)+1/64*A/a^4*b^4*(c*x^2+b*x+
a)^(3/2)+3/64*A/a^3*b^4*(c*x^2+b*x+a)^(1/2)-1/4*A/a/x^4*(c*x^2+b*x+a)^(5/2)-3/8*A*c^2/a^(1/2)*ln((b*x+2*a+2*(c
*x^2+b*x+a)^(1/2)*a^(1/2))/x)+B*c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/16*A/a^(3/2)*b^2*c*ln((b
*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-3/4*B/a^(1/2)*b*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+2/3*B
*c^2/a^2*(c*x^2+b*x+a)^(3/2)*x+B*c^2/a*(c*x^2+b*x+a)^(1/2)*x-1/32*A/a^3*b^2/x^2*(c*x^2+b*x+a)^(5/2)-1/8*A*c/a^
2/x^2*(c*x^2+b*x+a)^(5/2)+1/8*A/a^2*b/x^3*(c*x^2+b*x+a)^(5/2)+1/12*B/a^2*b/x^2*(c*x^2+b*x+a)^(5/2)+1/24*B/a^3*
b^2/x*(c*x^2+b*x+a)^(5/2)+7/12*B/a^2*b*c*(c*x^2+b*x+a)^(3/2)-1/64*A/a^4*b^3/x*(c*x^2+b*x+a)^(5/2)-5/32*A/a^3*b
^2*c*(c*x^2+b*x+a)^(3/2)-9/32*A/a^2*b^2*c*(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((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,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 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**5,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**5, x)

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