3.1210 \(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=107 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4}-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3} \]

[Out]

-1/6*(c*x^2+b*x+a)^(3/2)/c/d^4/(2*c*x+b)^3+1/16*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/d^4
-1/8*(c*x^2+b*x+a)^(1/2)/c^2/d^4/(2*c*x+b)

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Rubi [A]  time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {684, 621, 206} \[ -\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + b*x + c*x^2]/(8*c^2*d^4*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(6*c*d^4*(b + 2*c*x)^3) + ArcTanh[(b
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(16*c^(5/2)*d^4)

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 684

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[(b*p)/(d*e*(m + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1
), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^2} \, dx}{4 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\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 )}{8 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 95, normalized size = 0.89 \[ \frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{48 c^2 d^4 (b+2 c x)^3 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-3/2, -3/2, -1/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(48*c^2
*d^4*(b + 2*c*x)^3*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

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fricas [A]  time = 1.58, size = 345, normalized size = 3.22 \[ \left [\frac {3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (16 \, c^{3} x^{2} + 16 \, b c^{2} x + 3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{96 \, {\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}}, -\frac {3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (16 \, c^{3} x^{2} + 16 \, b c^{2} x + 3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{48 \, {\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*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*(16*c^3*x^2 + 16*b*c^2*x + 3*b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))
/(8*c^6*d^4*x^3 + 12*b*c^5*d^4*x^2 + 6*b^2*c^4*d^4*x + b^3*c^3*d^4), -1/48*(3*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^
2*c*x + b^3)*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*(16*c
^3*x^2 + 16*b*c^2*x + 3*b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a))/(8*c^6*d^4*x^3 + 12*b*c^5*d^4*x^2 + 6*b^2*c^4*
d^4*x + b^3*c^3*d^4)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%%{16,[4]%%%},[8,4,8,0]%%%}+%%%{%%%{-256,[5]%%%},[8,4,6,1]%%%}
+%%%{%%%{1536,[6]%%%},[8,4,4,2]%%%}+%%%{%%%{-4096,[7]%%%},[8,4,2,3]%%%}+%%%{%%%{4096,[8]%%%},[8,4,0,4]%%%}+%%%
{%%{[%%%{-64,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,4,9,0]%%%}+%%%{%%{[%%%{1024,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[7,4,7,1]%%%}+%%%{%%{[%%%{-6144,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,4,5,2]%%%}+%%%{%%{[%%%{16384,[6]%%%}
,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,4,3,3]%%%}+%%%{%%{[%%%{-16384,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,4,1,4]%%%}+
%%%{%%%{128,[3]%%%},[6,4,10,0]%%%}+%%%{%%%{-2112,[4]%%%},[6,4,8,1]%%%}+%%%{%%%{13312,[5]%%%},[6,4,6,2]%%%}+%%%
{%%%{-38912,[6]%%%},[6,4,4,3]%%%}+%%%{%%%{49152,[7]%%%},[6,4,2,4]%%%}+%%%{%%%{-16384,[8]%%%},[6,4,0,5]%%%}+%%%
{%%{[%%%{-160,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4,11,0]%%%}+%%%{%%{[%%%{2752,[3]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[5,4,9,1]%%%}+%%%{%%{[%%%{-18432,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4,7,2]%%%}+%%%{%%{[%%%{59392,[5]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4,5,3]%%%}+%%%{%%{[%%%{-90112,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4,3,4]%%
%}+%%%{%%{[%%%{49152,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4,1,5]%%%}+%%%{%%%{136,[2]%%%},[4,4,12,0]%%%}+%%%{%
%%{-2464,[3]%%%},[4,4,10,1]%%%}+%%%{%%%{17760,[4]%%%},[4,4,8,2]%%%}+%%%{%%%{-64000,[5]%%%},[4,4,6,3]%%%}+%%%{%
%%{117760,[6]%%%},[4,4,4,4]%%%}+%%%{%%%{-98304,[7]%%%},[4,4,2,5]%%%}+%%%{%%%{24576,[8]%%%},[4,4,0,6]%%%}+%%%{%
%{[%%%{-80,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,13,0]%%%}+%%%{%%{[%%%{1536,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%
%},[3,4,11,1]%%%}+%%%{%%{[%%%{-11968,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,9,2]%%%}+%%%{%%{[%%%{48128,[4]%%%
},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,7,3]%%%}+%%%{%%{[%%%{-104448,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,5,4]%%%
}+%%%{%%{[%%%{114688,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4,3,5]%%%}+%%%{%%{[%%%{-49152,[7]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[3,4,1,6]%%%}+%%%{%%%{32,[1]%%%},[2,4,14,0]%%%}+%%%{%%%{-656,[2]%%%},[2,4,12,1]%%%}+%%%{%%%{5568
,[3]%%%},[2,4,10,2]%%%}+%%%{%%%{-25152,[4]%%%},[2,4,8,3]%%%}+%%%{%%%{64512,[5]%%%},[2,4,6,4]%%%}+%%%{%%%{-9216
0,[6]%%%},[2,4,4,5]%%%}+%%%{%%%{65536,[7]%%%},[2,4,2,6]%%%}+%%%{%%%{-16384,[8]%%%},[2,4,0,7]%%%}+%%%{%%{[-8,0]
:[1,0,%%%{-1,[1]%%%}]%%},[1,4,15,0]%%%}+%%%{%%{[%%%{176,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,13,1]%%%}+%%%{
%%{[%%%{-1632,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,11,2]%%%}+%%%{%%{[%%%{8256,[3]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[1,4,9,3]%%%}+%%%{%%{[%%%{-24576,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,7,4]%%%}+%%%{%%{[%%%{43008,[5]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,5,5]%%%}+%%%{%%{[%%%{-40960,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,3,6]%%
%}+%%%{%%{[%%%{16384,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4,1,7]%%%}+%%%{1,[0,4,16,0]%%%}+%%%{%%%{-24,[1]%%%}
,[0,4,14,1]%%%}+%%%{%%%{248,[2]%%%},[0,4,12,2]%%%}+%%%{%%%{-1440,[3]%%%},[0,4,10,3]%%%}+%%%{%%%{5136,[4]%%%},[
0,4,8,4]%%%}+%%%{%%%{-11520,[5]%%%},[0,4,6,5]%%%}+%%%{%%%{15872,[6]%%%},[0,4,4,6]%%%}+%%%{%%%{-12288,[7]%%%},[
0,4,2,7]%%%}+%%%{%%%{4096,[8]%%%},[0,4,0,8]%%%} / %%%{%%%{16,[6]%%%},[8,0,0,0]%%%}+%%%{%%{poly1[%%%{-64,[5]%%%
},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0,1,0]%%%}+%%%{%%%{128,[5]%%%},[6,0,2,0]%%%}+%%%{%%%{-64,[6]%%%},[6,0,0,1]%%%}
+%%%{%%{poly1[%%%{-160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,3,0]%%%}+%%%{%%{[%%%{192,[5]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[5,0,1,1]%%%}+%%%{%%%{136,[4]%%%},[4,0,4,0]%%%}+%%%{%%%{-288,[5]%%%},[4,0,2,1]%%%}+%%%{%%%{96,[6]
%%%},[4,0,0,2]%%%}+%%%{%%{poly1[%%%{-80,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,5,0]%%%}+%%%{%%{[%%%{256,[4]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,1]%%%}+%%%{%%{poly1[%%%{-192,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,1,2]
%%%}+%%%{%%%{32,[3]%%%},[2,0,6,0]%%%}+%%%{%%%{-144,[4]%%%},[2,0,4,1]%%%}+%%%{%%%{192,[5]%%%},[2,0,2,2]%%%}+%%%
{%%%{-64,[6]%%%},[2,0,0,3]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,7,0]%%%}+%%%{%%{[%%
%{48,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,5,1]%%%}+%%%{%%{poly1[%%%{-96,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[1,0,3,2]%%%}+%%%{%%{[%%%{64,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,1,3]%%%}+%%%{%%%{1,[2]%%%},[0,0,8,0]%%%}+
%%%{%%%{-8,[3]%%%},[0,0,6,1]%%%}+%%%{%%%{24,[4]%%%},[0,0,4,2]%%%}+%%%{%%%{-32,[5]%%%},[0,0,2,3]%%%}+%%%{%%%{16
,[6]%%%},[0,0,0,4]%%%} Error: Bad Argument Value

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maple [B]  time = 0.06, size = 629, normalized size = 5.88 \[ \frac {a^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {c}\, d^{4}}-\frac {a \,b^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{2 \left (4 a c -b^{2}\right )^{2} c^{\frac {3}{2}} d^{4}}+\frac {b^{4} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{16 \left (4 a c -b^{2}\right )^{2} c^{\frac {5}{2}} d^{4}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a x}{\left (4 a c -b^{2}\right )^{2} d^{4}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{2} x}{4 \left (4 a c -b^{2}\right )^{2} c \,d^{4}}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a b}{2 \left (4 a c -b^{2}\right )^{2} c \,d^{4}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{3}}{8 \left (4 a c -b^{2}\right )^{2} c^{2} d^{4}}+\frac {2 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} x}{3 \left (4 a c -b^{2}\right )^{2} d^{4}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b}{3 \left (4 a c -b^{2}\right )^{2} c \,d^{4}}-\frac {2 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right ) c \,d^{4}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{12 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3} c^{3} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/d^4/c^3/(4*a*c-b^2)/(x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)-2/3/d^4/c/(4*a*c-b^2)^2/(x+1
/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+2/3/d^4/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3
/2)*x+1/3/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b+1/d^4/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c
+1/4*(4*a*c-b^2)/c)^(1/2)*x*a-1/4/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^2+1/2/d^4/
c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b*a-1/8/d^4/c^2/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(
4*a*c-b^2)/c)^(1/2)*b^3+1/d^4/c^(1/2)/(4*a*c-b^2)^2*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)
^(1/2))*a^2-1/2/d^4/c^(3/2)/(4*a*c-b^2)^2*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^
2*a+1/16/d^4/c^(5/2)/(4*a*c-b^2)^2*ln((x+1/2*b/c)*c^(1/2)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^4,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 = -0.01 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x)
+ Integral(b*x*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4)
, x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**
4*x**4), x))/d**4

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