3.30.59 \(\int \frac {(c+b x+a x^2)^{5/2}}{(c+b x)^2} \, dx\)

Optimal. Leaf size=362 \[ \frac {5 \left (2 a^{5/2} c^4-a^{3/2} b^2 c^3\right ) \log \left (\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{2 b^6}-\frac {5 \left (2 a^{5/2} c^4-a^{3/2} b^2 c^3\right ) \log \left (\sqrt {a} (b x+2 c)-b \sqrt {a x^2+b x+c}\right )}{2 b^6}+\frac {\sqrt {a x^2+b x+c} \left (48 a^3 b^4 x^4-80 a^3 b^3 c x^3+160 a^3 b^2 c^2 x^2-480 a^3 b c^3 x-960 a^3 c^4+136 a^2 b^5 x^3-64 a^2 b^4 c x^2+200 a^2 b^3 c^2 x+400 a^2 b^2 c^3+118 a b^6 x^2+146 a b^5 c x+28 a b^4 c^2+15 b^7 x+15 b^6 c\right )}{192 a b^5 (b x+c)}-\frac {5 \left (128 a^4 c^4-64 a^3 b^2 c^3+8 a b^6 c-b^8\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{128 a^{3/2} b^6} \]

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Rubi [A]  time = 0.32, antiderivative size = 277, normalized size of antiderivative = 0.77, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {732, 814, 843, 621, 206, 724} \begin {gather*} -\frac {5 a^{3/2} c^3 \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {x \left (b^2-2 a c\right )+b c}{2 \sqrt {a} c \sqrt {a x^2+b x+c}}\right )}{2 b^6}+\frac {5 \left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{64 a b^5}-\frac {5 \left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{128 a^{3/2} b^6}+\frac {5 \left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{24 b^3}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^6 - 4*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 + 2*a*b*(b^4 - 4*a*b^2*c + 16*a^2*c^2)*x)*Sqrt[c + b*x + a*x
^2])/(64*a*b^5) + (5*(7*b^2 - 8*a*c + 6*a*b*x)*(c + b*x + a*x^2)^(3/2))/(24*b^3) - (c + b*x + a*x^2)^(5/2)/(b*
(c + b*x)) - (5*(b^8 - 8*a*b^6*c + 64*a^3*b^2*c^3 - 128*a^4*c^4)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + b*x +
a*x^2])])/(128*a^(3/2)*b^6) - (5*a^(3/2)*c^3*(b^2 - 2*a*c)*ArcTanh[(b*c + (b^2 - 2*a*c)*x)/(2*Sqrt[a]*c*Sqrt[
c + b*x + a*x^2])])/(2*b^6)

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 732

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

Rule 814

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

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 {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx &=-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {5 \int \frac {(b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{c+b x} \, dx}{2 b}\\ &=\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {5 \int \frac {\left (-a b c \left (b^2+4 a c\right )-a \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{c+b x} \, dx}{16 a b^3}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {5 \int \frac {-\frac {1}{2} a b^5 c \left (b^2-8 a c\right )-\frac {1}{2} a \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) x}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{64 a^2 b^5}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {\left (5 a^2 c^4 \left (b^2-2 a c\right )\right ) \int \frac {1}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{2 b^6}-\frac {\left (5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right )\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{128 a b^6}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {\left (5 a^2 c^4 \left (b^2-2 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a c^2-x^2} \, dx,x,\frac {b c-\left (-b^2+2 a c\right ) x}{\sqrt {c+b x+a x^2}}\right )}{b^6}-\frac {\left (5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {b+2 a x}{\sqrt {c+b x+a x^2}}\right )}{64 a b^6}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{128 a^{3/2} b^6}-\frac {5 a^{3/2} c^3 \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b c+\left (b^2-2 a c\right ) x}{2 \sqrt {a} c \sqrt {c+b x+a x^2}}\right )}{2 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 271, normalized size = 0.75 \begin {gather*} \frac {5 \left (64 a^3 c^3 \left (2 a c-b^2\right ) \tanh ^{-1}\left (\frac {-2 a c x+b^2 x+b c}{2 \sqrt {a} c \sqrt {x (a x+b)+c}}\right )-\left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {x (a x+b)+c}}\right )+2 \sqrt {a} b \left (32 a^3 b c^2 x-64 a^3 c^3-8 a^2 b^3 c x+48 a^2 b^2 c^2+2 a b^5 x-4 a b^4 c+b^6\right ) \sqrt {x (a x+b)+c}\right )}{128 a^{3/2} b^6}+\frac {5 \left (6 a b x-8 a c+7 b^2\right ) (x (a x+b)+c)^{3/2}}{24 b^3}-\frac {(x (a x+b)+c)^{5/2}}{b (b x+c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(7*b^2 - 8*a*c + 6*a*b*x)*(c + x*(b + a*x))^(3/2))/(24*b^3) - (c + x*(b + a*x))^(5/2)/(b*(c + b*x)) + (5*(2
*Sqrt[a]*b*(b^6 - 4*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 + 2*a*b^5*x - 8*a^2*b^3*c*x + 32*a^3*b*c^2*x)*Sqrt[c
+ x*(b + a*x)] - (b^8 - 8*a*b^6*c + 64*a^3*b^2*c^3 - 128*a^4*c^4)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(
b + a*x)])] + 64*a^3*c^3*(-b^2 + 2*a*c)*ArcTanh[(b*c + b^2*x - 2*a*c*x)/(2*Sqrt[a]*c*Sqrt[c + x*(b + a*x)])]))
/(128*a^(3/2)*b^6)

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IntegrateAlgebraic [A]  time = 1.84, size = 366, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c+b x+a x^2} \left (15 b^6 c+28 a b^4 c^2+400 a^2 b^2 c^3-960 a^3 c^4+15 b^7 x+146 a b^5 c x+200 a^2 b^3 c^2 x-480 a^3 b c^3 x+118 a b^6 x^2-64 a^2 b^4 c x^2+160 a^3 b^2 c^2 x^2+136 a^2 b^5 x^3-80 a^3 b^3 c x^3+48 a^3 b^4 x^4\right )}{192 a b^5 (c+b x)}+\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{2 b^6}-\frac {5 \left (-b^8+8 a b^6 c-64 a^3 b^2 c^3+128 a^4 c^4\right ) \log \left (a b+2 a^2 x-2 a^{3/2} \sqrt {c+b x+a x^2}\right )}{128 a^{3/2} b^6}-\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+b x+a x^2}\right )}{2 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + b*x + a*x^2]*(15*b^6*c + 28*a*b^4*c^2 + 400*a^2*b^2*c^3 - 960*a^3*c^4 + 15*b^7*x + 146*a*b^5*c*x + 2
00*a^2*b^3*c^2*x - 480*a^3*b*c^3*x + 118*a*b^6*x^2 - 64*a^2*b^4*c*x^2 + 160*a^3*b^2*c^2*x^2 + 136*a^2*b^5*x^3
- 80*a^3*b^3*c*x^3 + 48*a^3*b^4*x^4))/(192*a*b^5*(c + b*x)) + (5*(-(a^(3/2)*b^2*c^3) + 2*a^(5/2)*c^4)*Log[-(Sq
rt[a]*x) + Sqrt[c + b*x + a*x^2]])/(2*b^6) - (5*(-b^8 + 8*a*b^6*c - 64*a^3*b^2*c^3 + 128*a^4*c^4)*Log[a*b + 2*
a^2*x - 2*a^(3/2)*Sqrt[c + b*x + a*x^2]])/(128*a^(3/2)*b^6) - (5*(-(a^(3/2)*b^2*c^3) + 2*a^(5/2)*c^4)*Log[Sqrt
[a]*(2*c + b*x) - b*Sqrt[c + b*x + a*x^2]])/(2*b^6)

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fricas [A]  time = 30.28, size = 854, normalized size = 2.36 \begin {gather*} \left [-\frac {15 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - 4 \, \sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {a} - b^{2} - 4 \, a c\right ) + 960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {a} \log \left (-\frac {2 \, b^{3} c x + b^{2} c^{2} + 4 \, a c^{3} + {\left (b^{4} - 4 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2} + 4 \, {\left (b c^{2} + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c} \sqrt {a}}{b^{2} x^{2} + 2 \, b c x + c^{2}}\right ) - 4 \, {\left (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{768 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}, -\frac {960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {-a} \arctan \left (-\frac {\sqrt {a x^{2} + b x + c} {\left (b c + {\left (b^{2} - 2 \, a c\right )} x\right )} \sqrt {-a}}{2 \, {\left (a^{2} c x^{2} + a b c x + a c^{2}\right )}}\right ) - 15 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 2 \, {\left (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{384 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(b^8*c - 8*a*b^6*c^2 + 64*a^3*b^2*c^4 - 128*a^4*c^5 + (b^9 - 8*a*b^7*c + 64*a^3*b^3*c^3 - 128*a^4*
b*c^4)*x)*sqrt(a)*log(-8*a^2*x^2 - 8*a*b*x - 4*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(a) - b^2 - 4*a*c) + 960*
(a^3*b^2*c^4 - 2*a^4*c^5 + (a^3*b^3*c^3 - 2*a^4*b*c^4)*x)*sqrt(a)*log(-(2*b^3*c*x + b^2*c^2 + 4*a*c^3 + (b^4 -
4*a*b^2*c + 8*a^2*c^2)*x^2 + 4*(b*c^2 + (b^2*c - 2*a*c^2)*x)*sqrt(a*x^2 + b*x + c)*sqrt(a))/(b^2*x^2 + 2*b*c*
x + c^2)) - 4*(48*a^4*b^5*x^4 + 15*a*b^7*c + 28*a^2*b^5*c^2 + 400*a^3*b^3*c^3 - 960*a^4*b*c^4 + 8*(17*a^3*b^6
- 10*a^4*b^4*c)*x^3 + 2*(59*a^2*b^7 - 32*a^3*b^5*c + 80*a^4*b^3*c^2)*x^2 + (15*a*b^8 + 146*a^2*b^6*c + 200*a^3
*b^4*c^2 - 480*a^4*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^2*b^7*x + a^2*b^6*c), -1/384*(960*(a^3*b^2*c^4 - 2*a^
4*c^5 + (a^3*b^3*c^3 - 2*a^4*b*c^4)*x)*sqrt(-a)*arctan(-1/2*sqrt(a*x^2 + b*x + c)*(b*c + (b^2 - 2*a*c)*x)*sqrt
(-a)/(a^2*c*x^2 + a*b*c*x + a*c^2)) - 15*(b^8*c - 8*a*b^6*c^2 + 64*a^3*b^2*c^4 - 128*a^4*c^5 + (b^9 - 8*a*b^7*
c + 64*a^3*b^3*c^3 - 128*a^4*b*c^4)*x)*sqrt(-a)*arctan(1/2*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(-a)/(a^2*x^2
+ a*b*x + a*c)) - 2*(48*a^4*b^5*x^4 + 15*a*b^7*c + 28*a^2*b^5*c^2 + 400*a^3*b^3*c^3 - 960*a^4*b*c^4 + 8*(17*a
^3*b^6 - 10*a^4*b^4*c)*x^3 + 2*(59*a^2*b^7 - 32*a^3*b^5*c + 80*a^4*b^3*c^2)*x^2 + (15*a*b^8 + 146*a^2*b^6*c +
200*a^3*b^4*c^2 - 480*a^4*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^2*b^7*x + a^2*b^6*c)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.43, size = 555, normalized size = 1.53

method	result	size
risch	\(\frac {\left (48 a^{3} x^{3} b^{3}-128 a^{3} b^{2} c \,x^{2}+136 a^{2} b^{4} x^{2}+288 a^{3} b \,c^{2} x -200 a^{2} b^{3} c x +118 a \,b^{5} x -768 a^{3} c^{3}+400 a^{2} b^{2} c^{2}+28 a \,b^{4} c +15 b^{6}\right ) \sqrt {a \,x^{2}+b x +c}}{192 a \,b^{5}}+\frac {5 a^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{4}}{b^{6}}-\frac {5 a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{3}}{2 b^{4}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c}{16 \sqrt {a}}-\frac {5 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{128 a^{\frac {3}{2}}}-\frac {a^{2} c^{4} \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{b^{6} \left (x +\frac {c}{b}\right )}+\frac {5 a^{3} c^{5} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{7} \sqrt {\frac {a \,c^{2}}{b^{2}}}}-\frac {5 a^{2} c^{4} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{2 b^{5} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\)	\(555\)
default	\(\frac {-\frac {b^{2} \left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {7}{2}}}{a \,c^{2} \left (x +\frac {c}{b}\right )}-\frac {5 \left (2 a c -b^{2}\right ) b \left (\frac {\left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {5}{2}}}{5}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{2 b}+\frac {a \,c^{2} \left (\frac {\left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{8 a^{\frac {3}{2}}}\right )}{2 b}+\frac {a \,c^{2} \left (\sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}-\frac {\left (2 a c -b^{2}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{2 b \sqrt {a}}-\frac {a \,c^{2} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{2} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\right )}{b^{2}}\right )}{b^{2}}\right )}{2 a \,c^{2}}+\frac {6 b^{2} \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {5}{2}}}{12 a}+\frac {5 \left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \left (\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{24 a}\right )}{c^{2}}}{b^{2}}\)	\(1378\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b*x+c)^(5/2)/(b*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/192/a*(48*a^3*b^3*x^3-128*a^3*b^2*c*x^2+136*a^2*b^4*x^2+288*a^3*b*c^2*x-200*a^2*b^3*c*x+118*a*b^5*x-768*a^3*
c^3+400*a^2*b^2*c^2+28*a*b^4*c+15*b^6)*(a*x^2+b*x+c)^(1/2)/b^5+5/b^6*a^(5/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x
+c)^(1/2))*c^4-5/2/b^4*a^(3/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))*c^3+5/16/a^(1/2)*ln((1/2*b+a*x)/a^(
1/2)+(a*x^2+b*x+c)^(1/2))*c-5/128*b^2/a^(3/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))-1/b^6*a^2*c^4/(x+c/b
)*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/b)+a*c^2/b^2)^(1/2)+5/b^7*a^3*c^5/(a*c^2/b^2)^(1/2)*ln((2*a*c^2/b^2-(2*a*c-b
^2)/b*(x+c/b)+2*(a*c^2/b^2)^(1/2)*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/b)+a*c^2/b^2)^(1/2))/(x+c/b))-5/2/b^5*a^2*c^
4/(a*c^2/b^2)^(1/2)*ln((2*a*c^2/b^2-(2*a*c-b^2)/b*(x+c/b)+2*(a*c^2/b^2)^(1/2)*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/
b)+a*c^2/b^2)^(1/2))/(x+c/b))

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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((a*x^2+b*x+c)^(5/2)/(b*x+c)^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 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\,x^2+b\,x+c\right )}^{5/2}}{{\left (c+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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