\(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^3} \, dx\) [2000]

Optimal result

Integrand size = 33, antiderivative size = 347 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {20 b^3 (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {2 b^5 (b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]

[Out]

-20*b^3*(-a*e+b*d)^3*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-1/2*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2+
6*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+15/2*b^4*(-a*e+b*d)^2*(e*x+d)^2*((b*x+a)^2)^(1/2)/e^7/(
b*x+a)-2*b^5*(-a*e+b*d)*(e*x+d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/4*b^6*(e*x+d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a
)+15*b^2*(-a*e+b*d)^4*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac {20 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)} \]

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-20*b^3*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d
+ e*x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) - (2*b^5*(b*d - a
*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(4*e^7*(a + b*x)) + (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^3} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^3} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {20 b^3 (b d-a e)^3}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^3}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^2}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^2}{e^6}+\frac {b^6 (d+e x)^3}{e^6}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {20 b^3 (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {2 b^5 (b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {\sqrt {(a+b x)^2} \left (-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+60 b^2 (b d-a e)^4 (d+e x)^2 \log (d+e x)\right )}{4 e^7 (a+b x) (d+e x)^2} \]

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(Sqrt[(a + b*x)^2]*(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3*b^3*e^3*(-
5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^
3 + e^4*x^4) + 4*a*b^5*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + b
^6*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + 60*b^2*(b
*d - a*e)^4*(d + e*x)^2*Log[d + e*x]))/(4*e^7*(a + b*x)*(d + e*x)^2)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (\frac {1}{4} b^{3} x^{4} e^{3}+2 x^{3} a \,b^{2} e^{3}-x^{3} b^{3} d \,e^{2}+\frac {15}{2} x^{2} a^{2} b \,e^{3}-9 x^{2} a \,b^{2} d \,e^{2}+3 x^{2} b^{3} d^{2} e +20 a^{3} e^{3} x -45 a^{2} b d \,e^{2} x +36 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-6 e^{5} a^{5} b +30 d \,e^{4} a^{4} b^{2}-60 d^{2} e^{3} a^{3} b^{3}+60 d^{3} e^{2} b^{4} a^{2}-30 d^{4} e \,b^{5} a +6 d^{5} b^{6}\right ) x -\frac {e^{6} a^{6}+6 b d \,e^{5} a^{5}-45 b^{2} d^{2} e^{4} a^{4}+100 b^{3} d^{3} e^{3} a^{3}-105 b^{4} d^{4} e^{2} a^{2}+54 b^{5} d^{5} e a -11 b^{6} d^{6}}{2 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{2}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\)	\(398\)
default	\(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (90 b^{2} d^{2} e^{4} a^{4}-200 b^{3} d^{3} e^{3} a^{3}+210 b^{4} d^{4} e^{2} a^{2}-108 b^{5} d^{5} e a -12 b d \,e^{5} a^{5}-24 a^{5} b \,e^{6} x -16 b^{6} d^{5} e x +8 a \,b^{5} e^{6} x^{5}-2 b^{6} d \,e^{5} x^{5}+30 a^{2} b^{4} e^{6} x^{4}+5 b^{6} d^{2} e^{4} x^{4}+80 a^{3} b^{3} e^{6} x^{3}-20 b^{6} d^{3} e^{3} x^{3}-68 b^{6} d^{4} e^{2} x^{2}-20 a \,b^{5} d \,e^{5} x^{4}-120 a^{2} b^{4} d \,e^{5} x^{3}+80 a \,b^{5} d^{2} e^{4} x^{3}+160 a^{3} b^{3} d \,e^{5} x^{2}-330 a^{2} b^{4} d^{2} e^{4} x^{2}+252 a \,b^{5} d^{3} e^{3} x^{2}+120 a^{4} b^{2} d \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-240 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}+120 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x -480 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x +720 \ln \left (e x +d \right ) a^{2} b^{4} d^{3} e^{3} x -480 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x +60 \ln \left (e x +d \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}-2 e^{6} a^{6}+22 b^{6} d^{6}-240 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-240 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-160 a^{3} b^{3} d^{2} e^{4} x +60 a^{2} b^{4} d^{3} e^{3} x +24 a \,b^{5} d^{4} e^{2} x +120 \ln \left (e x +d \right ) b^{6} d^{5} e x +b^{6} e^{6} x^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{4 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{2}}\)	\(669\)

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

((b*x+a)^2)^(1/2)/(b*x+a)*b^3/e^6*(1/4*b^3*x^4*e^3+2*x^3*a*b^2*e^3-x^3*b^3*d*e^2+15/2*x^2*a^2*b*e^3-9*x^2*a*b^
2*d*e^2+3*x^2*b^3*d^2*e+20*a^3*e^3*x-45*a^2*b*d*e^2*x+36*a*b^2*d^2*e*x-10*b^3*d^3*x)+((b*x+a)^2)^(1/2)/(b*x+a)
*((-6*a^5*b*e^5+30*a^4*b^2*d*e^4-60*a^3*b^3*d^2*e^3+60*a^2*b^4*d^3*e^2-30*a*b^5*d^4*e+6*b^6*d^5)*x-1/2*(a^6*e^
6+6*a^5*b*d*e^5-45*a^4*b^2*d^2*e^4+100*a^3*b^3*d^3*e^3-105*a^2*b^4*d^4*e^2+54*a*b^5*d^5*e-11*b^6*d^6)/e)/e^6/(
e*x+d)^2+15*((b*x+a)^2)^(1/2)/(b*x+a)*b^2/e^7*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*
ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (264) = 528\).

Time = 0.30 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \, {\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 2 \, {\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 200*a^3*b^3*d^3*e^3 + 90*a^4*b^2*d^2*e
^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b^6*d*e^5 - 4*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4
*e^6)*x^4 - 20*(b^6*d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3*e^6)*x^3 - 2*(34*b^6*d^4*e^2 - 126
*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 80*a^3*b^3*d*e^5)*x^2 - 4*(4*b^6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d
^3*e^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*
e^2 - 4*a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e
^5 + a^4*b^2*e^6)*x^2 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 - 4*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5
)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

Sympy [F]

\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(5/2)/(d + e*x)**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,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(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (264) = 528\).

Time = 0.28 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {15 \, {\left (b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 54 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 100 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {b^{6} e^{9} x^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, b^{6} d e^{8} x^{3} \mathrm {sgn}\left (b x + a\right ) + 8 \, a b^{5} e^{9} x^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{6} d^{2} e^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) - 36 \, a b^{5} d e^{8} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} e^{9} x^{2} \mathrm {sgn}\left (b x + a\right ) - 40 \, b^{6} d^{3} e^{6} x \mathrm {sgn}\left (b x + a\right ) + 144 \, a b^{5} d^{2} e^{7} x \mathrm {sgn}\left (b x + a\right ) - 180 \, a^{2} b^{4} d e^{8} x \mathrm {sgn}\left (b x + a\right ) + 80 \, a^{3} b^{3} e^{9} x \mathrm {sgn}\left (b x + a\right )}{4 \, e^{12}} \]

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

15*(b^6*d^4*sgn(b*x + a) - 4*a*b^5*d^3*e*sgn(b*x + a) + 6*a^2*b^4*d^2*e^2*sgn(b*x + a) - 4*a^3*b^3*d*e^3*sgn(b
*x + a) + a^4*b^2*e^4*sgn(b*x + a))*log(abs(e*x + d))/e^7 + 1/2*(11*b^6*d^6*sgn(b*x + a) - 54*a*b^5*d^5*e*sgn(
b*x + a) + 105*a^2*b^4*d^4*e^2*sgn(b*x + a) - 100*a^3*b^3*d^3*e^3*sgn(b*x + a) + 45*a^4*b^2*d^2*e^4*sgn(b*x +
a) - 6*a^5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a) + 12*(b^6*d^5*e*sgn(b*x + a) - 5*a*b^5*d^4*e^2*sgn(b*x
+ a) + 10*a^2*b^4*d^3*e^3*sgn(b*x + a) - 10*a^3*b^3*d^2*e^4*sgn(b*x + a) + 5*a^4*b^2*d*e^5*sgn(b*x + a) - a^5*
b*e^6*sgn(b*x + a))*x)/((e*x + d)^2*e^7) + 1/4*(b^6*e^9*x^4*sgn(b*x + a) - 4*b^6*d*e^8*x^3*sgn(b*x + a) + 8*a*
b^5*e^9*x^3*sgn(b*x + a) + 12*b^6*d^2*e^7*x^2*sgn(b*x + a) - 36*a*b^5*d*e^8*x^2*sgn(b*x + a) + 30*a^2*b^4*e^9*
x^2*sgn(b*x + a) - 40*b^6*d^3*e^6*x*sgn(b*x + a) + 144*a*b^5*d^2*e^7*x*sgn(b*x + a) - 180*a^2*b^4*d*e^8*x*sgn(
b*x + a) + 80*a^3*b^3*e^9*x*sgn(b*x + a))/e^12

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^3,x)

[Out]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^3, x)