3.4.29 \(\int \frac {(d+e x)^4}{(b x+c x^2)^{5/2}} \, dx\) [329]

Optimal. Leaf size=208 \[ -\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 832, 654, 634, 212} \begin {gather*} -\frac {2 (d+e x)^3 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 e \sqrt {b x+c x^2} (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{3 b^4 c^2}+\frac {4 (d+e x) \left (x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b c d^2 (4 c d-5 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(b*x + c*x^2)^(5/2),x]

[Out]

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

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 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 832

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^2 (d (4 c d-5 b e)-e (2 c d-b e) x)}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\frac {1}{2} b d e \left (8 c^2 d^2-12 b c d e+b^2 e^2\right )+\frac {1}{2} e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {e^4 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {\left (2 e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^2}\\ &=-\frac {2 (d+e x)^3 (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {4 (d+e x) \left (b c d^2 (4 c d-5 b e)+(2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {2 e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 195, normalized size = 0.94 \begin {gather*} -\frac {2 \left (\sqrt {c} \left (3 b^5 e^4 x^2-16 c^5 d^4 x^3+4 b^4 c e^4 x^3+8 b c^4 d^3 x^2 (-3 d+4 e x)-6 b^2 c^3 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )+b^3 c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )+3 b^4 e^4 x^{3/2} (b+c x)^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{3 b^4 c^{5/2} (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(Sqrt[c]*(3*b^5*e^4*x^2 - 16*c^5*d^4*x^3 + 4*b^4*c*e^4*x^3 + 8*b*c^4*d^3*x^2*(-3*d + 4*e*x) - 6*b^2*c^3*d^
2*x*(d^2 - 8*d*e*x + 2*e^2*x^2) + b^3*c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3)) + 3*b^4*e^4*x^(3/2)
*(b + c*x)^(3/2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]]))/(3*b^4*c^(5/2)*(x*(b + c*x))^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(190)=380\).
time = 0.49, size = 599, normalized size = 2.88

method	result	size
risch	\(-\frac {2 d^{3} \left (c x +b \right ) \left (12 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}+\frac {2 b \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e^{4}}{3 c^{4} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d \,e^{3}}{3 c^{3} \left (\frac {b}{c}+x \right )^{2}}+\frac {4 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{2} e^{2}}{b \,c^{2} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{3} e}{3 b^{2} c \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{4}}{3 b^{3} \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e^{4}}{3 c^{3} \left (\frac {b}{c}+x \right )}+\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d \,e^{3}}{3 b \,c^{2} \left (\frac {b}{c}+x \right )}+\frac {8 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{2} e^{2}}{b^{2} c \left (\frac {b}{c}+x \right )}-\frac {40 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{3} e}{3 b^{3} \left (\frac {b}{c}+x \right )}+\frac {16 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, d^{4}}{3 b^{4} \left (\frac {b}{c}+x \right )}\)	\(506\)
default	\(e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )}{2 c}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}}{c}\right )+4 d \,e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+6 d^{2} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+4 d^{3} e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+d^{4} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )\)	\(599\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (196) = 392\).
time = 0.29, size = 450, normalized size = 2.16 \begin {gather*} -\frac {1}{3} \, x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} e^{4} - \frac {4 \, c d^{4} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{4} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {8 \, d^{3} x e}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {64 \, c d^{3} x e}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d^{4}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{4}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {4 \, d x^{2} e^{3}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {8 \, d^{2} x e^{2}}{\sqrt {c x^{2} + b x} b^{2}} - \frac {4 \, d^{2} x e^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {32 \, d^{3} e}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {4 \, b d x e^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {8 \, d x e^{3}}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {4 \, d^{2} e^{2}}{\sqrt {c x^{2} + b x} b c} - \frac {4 \, x e^{4}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {4 \, d e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {2 \, \sqrt {c x^{2} + b x} e^{4}}{3 \, b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

-1/3*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2 + b*x)^(3/2)*c^2) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1/(sqrt(
c*x^2 + b*x)*c^2))*e^4 - 4/3*c*d^4*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d^4*x/(sqrt(c*x^2 + b*x)*b^4) + 8/3*
d^3*x*e/((c*x^2 + b*x)^(3/2)*b) - 64/3*c*d^3*x*e/(sqrt(c*x^2 + b*x)*b^3) - 2/3*d^4/((c*x^2 + b*x)^(3/2)*b) + 1
6/3*c*d^4/(sqrt(c*x^2 + b*x)*b^3) - 4*d*x^2*e^3/((c*x^2 + b*x)^(3/2)*c) + 8*d^2*x*e^2/(sqrt(c*x^2 + b*x)*b^2)
- 4*d^2*x*e^2/((c*x^2 + b*x)^(3/2)*c) - 32/3*d^3*e/(sqrt(c*x^2 + b*x)*b^2) - 4/3*b*d*x*e^3/((c*x^2 + b*x)^(3/2
)*c^2) + 8/3*d*x*e^3/(sqrt(c*x^2 + b*x)*b*c) + 4*d^2*e^2/(sqrt(c*x^2 + b*x)*b*c) - 4/3*x*e^4/(sqrt(c*x^2 + b*x
)*c^2) + 4/3*d*e^3/(sqrt(c*x^2 + b*x)*c^2) + e^4*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 2/3*sq
rt(c*x^2 + b*x)*e^4/(b*c^2)

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Fricas [A]
time = 1.71, size = 542, normalized size = 2.61 \begin {gather*} \left [\frac {3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {c} e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + 4 \, b^{3} c^{3} d x^{3} e^{3} - b^{3} c^{3} d^{4} - {\left (4 \, b^{4} c^{2} x^{3} + 3 \, b^{5} c x^{2}\right )} e^{4} + 6 \, {\left (2 \, b^{2} c^{4} d^{2} x^{3} + 3 \, b^{3} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, b^{3} c^{3} d^{3} x\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) e^{4} - {\left (16 \, c^{6} d^{4} x^{3} + 24 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + 4 \, b^{3} c^{3} d x^{3} e^{3} - b^{3} c^{3} d^{4} - {\left (4 \, b^{4} c^{2} x^{3} + 3 \, b^{5} c x^{2}\right )} e^{4} + 6 \, {\left (2 \, b^{2} c^{4} d^{2} x^{3} + 3 \, b^{3} c^{3} d^{2} x^{2}\right )} e^{2} - 4 \, {\left (8 \, b c^{5} d^{3} x^{3} + 12 \, b^{2} c^{4} d^{3} x^{2} + 3 \, b^{3} c^{3} d^{3} x\right )} e\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*sqrt(c)*e^4*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(16
*c^6*d^4*x^3 + 24*b*c^5*d^4*x^2 + 6*b^2*c^4*d^4*x + 4*b^3*c^3*d*x^3*e^3 - b^3*c^3*d^4 - (4*b^4*c^2*x^3 + 3*b^5
*c*x^2)*e^4 + 6*(2*b^2*c^4*d^2*x^3 + 3*b^3*c^3*d^2*x^2)*e^2 - 4*(8*b*c^5*d^3*x^3 + 12*b^2*c^4*d^3*x^2 + 3*b^3*
c^3*d^3*x)*e)*sqrt(c*x^2 + b*x))/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2), -2/3*(3*(b^4*c^2*x^4 + 2*b^5*c*x
^3 + b^6*x^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))*e^4 - (16*c^6*d^4*x^3 + 24*b*c^5*d^4*x^2 + 6*b
^2*c^4*d^4*x + 4*b^3*c^3*d*x^3*e^3 - b^3*c^3*d^4 - (4*b^4*c^2*x^3 + 3*b^5*c*x^2)*e^4 + 6*(2*b^2*c^4*d^2*x^3 +
3*b^3*c^3*d^2*x^2)*e^2 - 4*(8*b*c^5*d^3*x^3 + 12*b^2*c^4*d^3*x^2 + 3*b^3*c^3*d^3*x)*e)*sqrt(c*x^2 + b*x))/(b^4
*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d + e*x)**4/(x*(b + c*x))**(5/2), x)

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Giac [A]
time = 1.36, size = 209, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {d^{4}}{b} - {\left (x {\left (\frac {4 \, {\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x}{b^{4} c^{2}} + \frac {3 \, {\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - b^{5} e^{4}\right )}}{b^{4} c^{2}}\right )} + \frac {6 \, {\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} - \frac {e^{4} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

-2/3*(d^4/b - (x*(4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + b^3*c^2*d*e^3 - b^4*c*e^4)*x/(b^4*c^2) +
3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e + 6*b^3*c^2*d^2*e^2 - b^5*e^4)/(b^4*c^2)) + 6*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e
)/(b^4*c^2))*x)/(c*x^2 + b*x)^(3/2) - e^4*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^4/(b*x + c*x^2)^(5/2), x)

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