\(\int \frac {(b d+2 c d x)^4}{(a+b x+c x^2)^{3/2}} \, dx\) [1242]

Optimal result

Integrand size = 26, antiderivative size = 102 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}+6 \sqrt {c} \left (b^2-4 a c\right ) d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

[Out]

6*(-4*a*c+b^2)*d^4*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)-2*d^4*(2*c*x+b)^3/(c*x^2+b*x+a)^
(1/2)+12*c*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {700, 706, 635, 212} \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=6 \sqrt {c} d^4 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}} \]

[In]

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

[Out]

(-2*d^4*(b + 2*c*x)^3)/Sqrt[a + b*x + c*x^2] + 12*c*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 6*Sqrt[c]*(b^2 - 4
*a*c)*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 700

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*(d + e*x)^(m - 1)*(
(a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Dist[d*e*((m - 1)/(b*(p + 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] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+\left (12 c d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\left (6 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right ) d^4\right ) \text {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 {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}+6 \sqrt {c} \left (b^2-4 a c\right ) d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.91 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=d^4 \left (-\frac {2 (b+2 c x) \left (b^2-2 b c x-2 c \left (3 a+c x^2\right )\right )}{\sqrt {a+x (b+c x)}}+12 \sqrt {c} \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right ) \]

[In]

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

[Out]

d^4*((-2*(b + 2*c*x)*(b^2 - 2*b*c*x - 2*c*(3*a + c*x^2)))/Sqrt[a + x*(b + c*x)] + 12*Sqrt[c]*(b^2 - 4*a*c)*Arc
Tanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(335\) vs. \(2(90)=180\).

Time = 3.04 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.29

method	result	size
risch	\(4 c \,d^{4} \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}+\left (\frac {2 b^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {16 a^{2} c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 a \,b^{2} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\left (24 c^{3} a -6 b^{2} c^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )-\left (24 b \,c^{2} a -6 b^{3} c \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\right ) d^{4}\)	\(336\)
default	\(d^{4} \left (\frac {2 b^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+16 c^{4} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+8 b^{3} c \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+32 c^{3} b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+24 b^{2} c^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\right )\)	\(757\)

[In]

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

[Out]

4*c*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)+(2*b^4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-16*a^2*c^2*(2*c*x+b)/(4
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-4*a*b^2*c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-(24*a*c^3-6*b^2*c^2)*(-x/c/(
c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-(24*a*b*c^2-6*b^3*c)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)))*d^4

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.50 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{4} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 2 \, {\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x - {\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{c x^{2} + b x + a}, -\frac {2 \, {\left (3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{4} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - {\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x - {\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \]

[In]

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

[Out]

[-(3*((b^2*c - 4*a*c^2)*d^4*x^2 + (b^3 - 4*a*b*c)*d^4*x + (a*b^2 - 4*a^2*c)*d^4)*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) - 2*(4*c^3*d^4*x^3 + 6*b*c^2*d^4*x^2 + 12*a*c
^2*d^4*x - (b^3 - 6*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/(c*x^2 + b*x + a), -2*(3*((b^2*c - 4*a*c^2)*d^4*x^2 + (
b^3 - 4*a*b*c)*d^4*x + (a*b^2 - 4*a^2*c)*d^4)*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)) - (4*c^3*d^4*x^3 + 6*b*c^2*d^4*x^2 + 12*a*c^2*d^4*x - (b^3 - 6*a*b*c)*d^4)*sqrt(c*x^2
+ b*x + a))/(c*x^2 + b*x + a)]

Sympy [F]

\[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=d^{4} \left (\int \frac {b^{4}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {16 c^{4} x^{4}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {32 b c^{3} x^{3}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {24 b^{2} c^{2} x^{2}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {8 b^{3} c x}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (90) = 180\).

Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.42 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {6 \, {\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{\sqrt {c}} + \frac {2 \, {\left (2 \, {\left ({\left (\frac {2 \, {\left (b^{2} c^{5} d^{4} - 4 \, a c^{6} d^{4}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, {\left (b^{3} c^{4} d^{4} - 4 \, a b c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {6 \, {\left (a b^{2} c^{4} d^{4} - 4 \, a^{2} c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac {b^{5} c^{2} d^{4} - 10 \, a b^{3} c^{3} d^{4} + 24 \, a^{2} b c^{4} d^{4}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt {c x^{2} + b x + a}} \]

[In]

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

[Out]

-6*(b^2*c*d^4 - 4*a*c^2*d^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 2*(2*((2*(b
^2*c^5*d^4 - 4*a*c^6*d^4)*x/(b^2*c^2 - 4*a*c^3) + 3*(b^3*c^4*d^4 - 4*a*b*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x + 6*(
a*b^2*c^4*d^4 - 4*a^2*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x - (b^5*c^2*d^4 - 10*a*b^3*c^3*d^4 + 24*a^2*b*c^4*d^4)/(b
^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a)

Mupad [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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