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

Optimal. Leaf size=334 \[ -\frac {a^{5/2} c^5 \log \left (\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{b^6}+\frac {a^{5/2} c^5 \log \left (-b \sqrt {a x^2+b x+c}+\sqrt {a} b x+2 \sqrt {a} c\right )}{b^6}+\frac {\left (256 a^5 c^5-80 a^2 b^6 c^2+30 a b^8 c-3 b^{10}\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{256 a^{5/2} b^6}+\frac {\sqrt {a x^2+b x+c} \left (384 a^4 b^4 x^4-480 a^4 b^3 c x^3+640 a^4 b^2 c^2 x^2-960 a^4 b c^3 x+1920 a^4 c^4+1008 a^3 b^5 x^3+48 a^3 b^4 c x^2-80 a^3 b^3 c^2 x+160 a^3 b^2 c^3+744 a^2 b^6 x^2+1308 a^2 b^5 c x+24 a^2 b^4 c^2+30 a b^7 x+390 a b^6 c-45 b^8\right )}{1920 a^2 b^5} \]

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Rubi [A]  time = 0.33, antiderivative size = 309, normalized size of antiderivative = 0.93, 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 = {734, 814, 843, 621, 206, 724} \begin {gather*} -\frac {a^{5/2} c^5 \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 )}{b^6}+\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{48 a b^3}+\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{256 a^{5/2} b^6}-\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{128 a^2 b^5}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((3*b^8 - 18*a*b^6*c + 8*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 2*a*b*(b^2 - 2*a*c)*(3*b^4 - 12*a
*b^2*c - 16*a^2*c^2)*x)*Sqrt[c + b*x + a*x^2])/(a^2*b^5) + ((3*b^4 - 6*a*b^2*c + 16*a^2*c^2 + 6*a*b*(b^2 - 2*a
*c)*x)*(c + b*x + a*x^2)^(3/2))/(48*a*b^3) + (c + b*x + a*x^2)^(5/2)/(5*b) + ((3*b^10 - 30*a*b^8*c + 80*a^2*b^
6*c^2 - 256*a^5*c^5)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + b*x + a*x^2])])/(256*a^(5/2)*b^6) - (a^(5/2)*c^5*
ArcTanh[(b*c + (b^2 - 2*a*c)*x)/(2*Sqrt[a]*c*Sqrt[c + b*x + a*x^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 734

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 + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[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} \, dx &=\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\int \frac {\left (-b c-\left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{c+b x} \, dx}{2 b}\\ &=\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\int \frac {\left (-\frac {1}{2} b c \left (3 b^4-18 a b^2 c+8 a^2 c^2\right )-\frac {1}{2} \left (b^2-2 a c\right ) \left (3 b^4-12 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 {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\int \frac {-\frac {1}{4} b^5 c \left (3 b^4-30 a b^2 c+80 a^2 c^2\right )-\frac {1}{4} \left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) x}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{64 a^2 b^5}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\left (a^3 c^6\right ) \int \frac {1}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{b^6}-\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{256 a^2 b^6}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\left (2 a^3 c^6\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 (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\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 )}{128 a^2 b^6}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{256 a^{5/2} b^6}-\frac {a^{5/2} c^5 \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 )}{b^6}\\ \end {align*}

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Mathematica [A]  time = 0.53, size = 275, normalized size = 0.82 \begin {gather*} \frac {-3840 a^5 c^5 \tanh ^{-1}\left (\frac {-2 a c x+b^2 x+b c}{2 \sqrt {a} c \sqrt {x (a x+b)+c}}\right )+15 \left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {x (a x+b)+c}}\right )+2 \sqrt {a} b \sqrt {x (a x+b)+c} \left (-960 a^4 b c^3 x+1920 a^4 c^4-80 a^3 b^3 c x \left (6 a x^2+c\right )+160 a^3 b^2 c^2 \left (4 a x^2+c\right )+12 a^2 b^5 x \left (84 a x^2+109 c\right )+24 a^2 b^4 \left (16 a^2 x^4+2 a c x^2+c^2\right )+30 a b^7 x+6 a b^6 \left (124 a x^2+65 c\right )-45 b^8\right )}{3840 a^{5/2} b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a]*b*Sqrt[c + x*(b + a*x)]*(-45*b^8 + 1920*a^4*c^4 + 30*a*b^7*x - 960*a^4*b*c^3*x + 160*a^3*b^2*c^2*(c
+ 4*a*x^2) - 80*a^3*b^3*c*x*(c + 6*a*x^2) + 12*a^2*b^5*x*(109*c + 84*a*x^2) + 6*a*b^6*(65*c + 124*a*x^2) + 24
*a^2*b^4*(c^2 + 2*a*c*x^2 + 16*a^2*x^4)) + 15*(3*b^10 - 30*a*b^8*c + 80*a^2*b^6*c^2 - 256*a^5*c^5)*ArcTanh[(b
+ 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])] - 3840*a^5*c^5*ArcTanh[(b*c + b^2*x - 2*a*c*x)/(2*Sqrt[a]*c*Sqrt[c
+ x*(b + a*x)])])/(3840*a^(5/2)*b^6)

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IntegrateAlgebraic [A]  time = 1.71, size = 334, normalized size = 1.00 \begin {gather*} \frac {\sqrt {c+b x+a x^2} \left (-45 b^8+390 a b^6 c+24 a^2 b^4 c^2+160 a^3 b^2 c^3+1920 a^4 c^4+30 a b^7 x+1308 a^2 b^5 c x-80 a^3 b^3 c^2 x-960 a^4 b c^3 x+744 a^2 b^6 x^2+48 a^3 b^4 c x^2+640 a^4 b^2 c^2 x^2+1008 a^3 b^5 x^3-480 a^4 b^3 c x^3+384 a^4 b^4 x^4\right )}{1920 a^2 b^5}-\frac {a^{5/2} c^5 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{b^6}+\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{256 a^{5/2} b^6}+\frac {a^{5/2} c^5 \log \left (2 \sqrt {a} c+\sqrt {a} b x-b \sqrt {c+b x+a x^2}\right )}{b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + b*x + a*x^2]*(-45*b^8 + 390*a*b^6*c + 24*a^2*b^4*c^2 + 160*a^3*b^2*c^3 + 1920*a^4*c^4 + 30*a*b^7*x +
1308*a^2*b^5*c*x - 80*a^3*b^3*c^2*x - 960*a^4*b*c^3*x + 744*a^2*b^6*x^2 + 48*a^3*b^4*c*x^2 + 640*a^4*b^2*c^2*
x^2 + 1008*a^3*b^5*x^3 - 480*a^4*b^3*c*x^3 + 384*a^4*b^4*x^4))/(1920*a^2*b^5) - (a^(5/2)*c^5*Log[-(Sqrt[a]*x)
+ Sqrt[c + b*x + a*x^2]])/b^6 + ((-3*b^10 + 30*a*b^8*c - 80*a^2*b^6*c^2 + 256*a^5*c^5)*Log[b + 2*a*x - 2*Sqrt[
a]*Sqrt[c + b*x + a*x^2]])/(256*a^(5/2)*b^6) + (a^(5/2)*c^5*Log[2*Sqrt[a]*c + Sqrt[a]*b*x - b*Sqrt[c + b*x + a
*x^2]])/b^6

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fricas [A]  time = 84.85, size = 707, normalized size = 2.12 \begin {gather*} \left [\frac {3840 \, a^{\frac {11}{2}} c^{5} \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 ) - 15 \, {\left (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 ) + 4 \, {\left (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{7680 \, a^{3} b^{6}}, -\frac {3840 \, \sqrt {-a} a^{5} c^{5} \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 (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{3840 \, a^{3} b^{6}}\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),x, algorithm="fricas")

[Out]

[1/7680*(3840*a^(11/2)*c^5*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)) - 15*(3*b^10 - 30*a*b^8*c + 8
0*a^2*b^6*c^2 - 256*a^5*c^5)*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) + 4*(384*a^5*b^5*x^4 - 45*a*b^9 + 390*a^2*b^7*c + 24*a^3*b^5*c^2 + 160*a^4*b^3*c^3 + 1920*a^5*b*c
^4 + 48*(21*a^4*b^6 - 10*a^5*b^4*c)*x^3 + 8*(93*a^3*b^7 + 6*a^4*b^5*c + 80*a^5*b^3*c^2)*x^2 + 2*(15*a^2*b^8 +
654*a^3*b^6*c - 40*a^4*b^4*c^2 - 480*a^5*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^3*b^6), -1/3840*(3840*sqrt(-a)*
a^5*c^5*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
*(3*b^10 - 30*a*b^8*c + 80*a^2*b^6*c^2 - 256*a^5*c^5)*sqrt(-a)*arctan(1/2*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sq
rt(-a)/(a^2*x^2 + a*b*x + a*c)) - 2*(384*a^5*b^5*x^4 - 45*a*b^9 + 390*a^2*b^7*c + 24*a^3*b^5*c^2 + 160*a^4*b^3
*c^3 + 1920*a^5*b*c^4 + 48*(21*a^4*b^6 - 10*a^5*b^4*c)*x^3 + 8*(93*a^3*b^7 + 6*a^4*b^5*c + 80*a^5*b^3*c^2)*x^2
+ 2*(15*a^2*b^8 + 654*a^3*b^6*c - 40*a^4*b^4*c^2 - 480*a^5*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^3*b^6)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [A]  time = 0.45, size = 431, normalized size = 1.29

method	result	size
risch	\(\frac {\sqrt {a \,x^{2}+b x +c}\, \left (384 a^{4} b^{4} x^{4}-480 a^{4} b^{3} c \,x^{3}+1008 a^{3} b^{5} x^{3}+640 a^{4} b^{2} c^{2} x^{2}+48 a^{3} b^{4} c \,x^{2}+744 a^{2} b^{6} x^{2}-960 a^{4} b \,c^{3} x -80 a^{3} b^{3} c^{2} x +1308 a^{2} b^{5} c x +30 a \,b^{7} x +1920 a^{4} c^{4}+160 a^{3} b^{2} c^{3}+24 a^{2} b^{4} c^{2}+390 b^{6} c a -45 b^{8}\right )}{1920 a^{2} b^{5}}-\frac {a^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{5}}{b^{6}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{2}}{16 \sqrt {a}}-\frac {15 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c}{128 a^{\frac {3}{2}}}+\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{256 a^{\frac {5}{2}}}-\frac {a^{3} c^{6} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{7} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\)	\(431\)
default	\(\frac {\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}}}{b}\)	\(881\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*(a*x^2+b*x+c)^(1/2)*(384*a^4*b^4*x^4-480*a^4*b^3*c*x^3+1008*a^3*b^5*x^3+640*a^4*b^2*c^2*x^2+48*a^3*b^4*
c*x^2+744*a^2*b^6*x^2-960*a^4*b*c^3*x-80*a^3*b^3*c^2*x+1308*a^2*b^5*c*x+30*a*b^7*x+1920*a^4*c^4+160*a^3*b^2*c^
3+24*a^2*b^4*c^2+390*a*b^6*c-45*b^8)/a^2/b^5-a^(5/2)/b^6*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))*c^5+5/16/
a^(1/2)*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))*c^2-15/128/a^(3/2)*b^2*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c
)^(1/2))*c+3/256/a^(5/2)*b^4*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))-a^3/b^7*c^6/(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),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}}{c+b\,x} \,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),x)

[Out]

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

[Out]

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

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