3.40 \(\int \frac{(a x^2+b x^3+c x^4)^{3/2}}{x} \, dx\)

Optimal. Leaf size=288 \[ \frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]

[Out]

((35*b^4 - 216*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(3840*c^3) - (b*(105*b^4 - 760*a*b^2*c + 12
96*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(7680*c^4*x) - (x*(b*(7*b^2 + 12*a*c) + 6*c*(7*b^2 - 20*a*c)*x)*Sqrt[
a*x^2 + b*x^3 + c*x^4])/(960*c^2) + ((3*b + 10*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(60*c*x) + ((b^2 - 4*a*c)^2
*(7*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2)
*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.519235, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1919, 1934, 1949, 12, 1914, 621, 206} \[ \frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x]

[Out]

((35*b^4 - 216*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(3840*c^3) - (b*(105*b^4 - 760*a*b^2*c + 12
96*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(7680*c^4*x) - (x*(b*(7*b^2 + 12*a*c) + 6*c*(7*b^2 - 20*a*c)*x)*Sqrt[
a*x^2 + b*x^3 + c*x^4])/(960*c^2) + ((3*b + 10*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(60*c*x) + ((b^2 - 4*a*c)^2
*(7*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2)
*Sqrt[a*x^2 + b*x^3 + c*x^4])

Rule 1919

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[(x^(m - n + q
+ 1)*(b*(n - q)*p + c*(m + p*q + (n - q)*(2*p - 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(c*(m +
p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1)), x] + Dist[((n - q)*p)/(c*(m + p*(2*n - q) + 1)*(m + p*q +
(n - q)*(2*p - 1) + 1)), Int[x^(m - (n - 2*q))*Simp[-(a*b*(m + p*q - n + q + 1)) + (2*a*c*(m + p*q + (n - q)*
(2*p - 1) + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x
], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ
[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, n - q] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*
q + (n - q)*(2*p - 1) + 1, 0]

Rule 1934

Int[((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp
[(x*(b*B*(n - q)*p + A*c*(p*q + (n - q)*(2*p + 1) + 1) + B*c*(p*(2*n - q) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x
^(2*n - q))^p)/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)), x] + Dist[((n - q)*p)/(c*(p*(2*n - q) + 1)
*(p*q + (n - q)*(2*p + 1) + 1)), Int[x^q*(2*a*A*c*(p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(p*q + 1) + (2*a*B*c*(
p*(2*n - q) + 1) + A*b*c*(p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(p*q + (n - q)*p + 1))*x^(n - q))*(a*x^q + b*x^
n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] &&
!IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + (n - q)*(2*p + 1) + 1
, 0]

Rule 1949

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> Simp[(B*x^(m - n + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(c*(m + p*q + (n - q)*(2*p + 1) + 1)),
x] - Dist[1/(c*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m - n + q)*Simp[a*B*(m + p*q - n + q + 1) + (b*B*(m
+ p*q + (n - q)*p + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^
p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c
, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (
n - q)*(2*p + 1) + 1, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1914

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[(x^(q/2)*Sqrt[a
+ b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) +
c*x^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && ((EqQ[m, 1] &&
EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q,
1]))

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 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])

Rubi steps

\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx &=\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \left (-2 a b+\frac{1}{2} \left (-7 b^2+20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4} \, dx}{20 c}\\ &=-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \frac{x^2 \left (-16 a^2 b c-a b \left (-7 b^2+20 a c\right )+\left (-8 a b^2 c-\frac{5}{4} b^2 \left (-7 b^2+20 a c\right )+3 a c \left (-7 b^2+20 a c\right )\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{480 c^2}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac{\int \frac{x \left (\frac{1}{4} a \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )+\frac{1}{8} b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{960 c^3}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \frac{15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x}{16 \sqrt{a x^2+b x^3+c x^4}} \, dx}{960 c^4}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{1024 c^4}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^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 )}{512 c^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}

Mathematica [A]  time = 0.242008, size = 180, normalized size = 0.62 \[ \frac{\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac{\left (7 b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{512 c^{7/2} (a+x (b+c x))^{3/2}}+x (a+x (b+c x))-\frac{7 b (a+x (b+c x))}{10 c}\right )}{6 c x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x]

[Out]

((x^2*(a + x*(b + c*x)))^(3/2)*((-7*b*(a + x*(b + c*x)))/(10*c) + x*(a + x*(b + c*x)) + ((7*b^2 - 4*a*c)*(2*Sq
rt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(512*c^(7/2)*(a + x*(b + c*x))^(3/2))))/(6*c*x^3)

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Maple [A]  time = 0.009, size = 431, normalized size = 1.5 \begin{align*}{\frac{1}{15360\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2560\,x \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}-1792\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}b-640\,{c}^{9/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}xa+1120\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}x{b}^{2}-320\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}ab+560\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{b}^{3}-960\,{c}^{9/2}\sqrt{c{x}^{2}+bx+a}x{a}^{2}+1920\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}xa{b}^{2}-420\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}x{b}^{4}-480\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}{a}^{2}b+960\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}a{b}^{3}-210\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{5}-960\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}{c}^{4}+2160\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{2}{c}^{3}-900\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{4}{c}^{2}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{6}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)^(3/2)/x,x)

[Out]

1/15360*(c*x^4+b*x^3+a*x^2)^(3/2)*(2560*x*(c*x^2+b*x+a)^(5/2)*c^(9/2)-1792*c^(7/2)*(c*x^2+b*x+a)^(5/2)*b-640*c
^(9/2)*(c*x^2+b*x+a)^(3/2)*x*a+1120*c^(7/2)*(c*x^2+b*x+a)^(3/2)*x*b^2-320*c^(7/2)*(c*x^2+b*x+a)^(3/2)*a*b+560*
c^(5/2)*(c*x^2+b*x+a)^(3/2)*b^3-960*c^(9/2)*(c*x^2+b*x+a)^(1/2)*x*a^2+1920*c^(7/2)*(c*x^2+b*x+a)^(1/2)*x*a*b^2
-420*c^(5/2)*(c*x^2+b*x+a)^(1/2)*x*b^4-480*c^(7/2)*(c*x^2+b*x+a)^(1/2)*a^2*b+960*c^(5/2)*(c*x^2+b*x+a)^(1/2)*a
*b^3-210*c^(3/2)*(c*x^2+b*x+a)^(1/2)*b^5-960*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^3*c^4+2
160*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^2*b^2*c^3-900*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1
/2)+2*c*x+b)/c^(1/2))*a*b^4*c^2+105*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^6*c)/x^3/(c*x^2+
b*x+a)^(3/2)/c^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x, x)

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Fricas [A]  time = 1.83513, size = 1099, normalized size = 3.82 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{30720 \, c^{5} x}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{15360 \, c^{5} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="fricas")

[Out]

[-1/30720*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)*x*log(-(8*c^2*x^3 + 8*b*c*x^2 - 4*sq
rt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(c) + (b^2 + 4*a*c)*x)/x) - 4*(1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b
^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 16*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(3
5*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x), -1/15360*(15*(7*b^6 - 60*a*b
^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-c)*x*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c
^2*x^3 + b*c*x^2 + a*c*x)) - 2*(1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 1
6*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*
x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**3+a*x**2)**(3/2)/x,x)

[Out]

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

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Giac [A]  time = 1.32493, size = 493, normalized size = 1.71 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x \mathrm{sgn}\left (x\right ) + 13 \, b \mathrm{sgn}\left (x\right )\right )} x + \frac{3 \, b^{2} c^{4} \mathrm{sgn}\left (x\right ) + 140 \, a c^{5} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x - \frac{7 \, b^{3} c^{3} \mathrm{sgn}\left (x\right ) - 36 \, a b c^{4} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x + \frac{35 \, b^{4} c^{2} \mathrm{sgn}\left (x\right ) - 216 \, a b^{2} c^{3} \mathrm{sgn}\left (x\right ) + 240 \, a^{2} c^{4} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x - \frac{105 \, b^{5} c \mathrm{sgn}\left (x\right ) - 760 \, a b^{3} c^{2} \mathrm{sgn}\left (x\right ) + 1296 \, a^{2} b c^{3} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} - \frac{{\left (7 \, b^{6} \mathrm{sgn}\left (x\right ) - 60 \, a b^{4} c \mathrm{sgn}\left (x\right ) + 144 \, a^{2} b^{2} c^{2} \mathrm{sgn}\left (x\right ) - 64 \, a^{3} c^{3} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} + \frac{{\left (105 \, b^{6} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 900 \, a b^{4} c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2160 \, a^{2} b^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 960 \, a^{3} c^{3} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 210 \, \sqrt{a} b^{5} \sqrt{c} - 1520 \, a^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} + 2592 \, a^{\frac{5}{2}} b c^{\frac{5}{2}}\right )} \mathrm{sgn}\left (x\right )}{15360 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="giac")

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*sgn(x) + 13*b*sgn(x))*x + (3*b^2*c^4*sgn(x) + 140*a*c^5*sgn(x
))/c^5)*x - (7*b^3*c^3*sgn(x) - 36*a*b*c^4*sgn(x))/c^5)*x + (35*b^4*c^2*sgn(x) - 216*a*b^2*c^3*sgn(x) + 240*a^
2*c^4*sgn(x))/c^5)*x - (105*b^5*c*sgn(x) - 760*a*b^3*c^2*sgn(x) + 1296*a^2*b*c^3*sgn(x))/c^5) - 1/1024*(7*b^6*
sgn(x) - 60*a*b^4*c*sgn(x) + 144*a^2*b^2*c^2*sgn(x) - 64*a^3*c^3*sgn(x))*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c) - b))/c^(9/2) + 1/15360*(105*b^6*log(abs(-b + 2*sqrt(a)*sqrt(c))) - 900*a*b^4*c*log(abs(-b +
2*sqrt(a)*sqrt(c))) + 2160*a^2*b^2*c^2*log(abs(-b + 2*sqrt(a)*sqrt(c))) - 960*a^3*c^3*log(abs(-b + 2*sqrt(a)*
sqrt(c))) + 210*sqrt(a)*b^5*sqrt(c) - 1520*a^(3/2)*b^3*c^(3/2) + 2592*a^(5/2)*b*c^(5/2))*sgn(x)/c^(9/2)