3.2000 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=347 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac{20 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)} \]

[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))

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Rubi [A]  time = 0.258594, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac{20 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[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 770

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]

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 43

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])

Rubi steps

\begin{align*} \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^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]  time = 0.153159, size = 321, normalized size = 0.93 \[ \frac{\sqrt{(a+b x)^2} \left (30 a^2 b^4 e^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+40 a^3 b^3 e^3 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+30 a^4 b^2 d e^4 (3 d+4 e x)-12 a^5 b e^5 (d+2 e x)-2 a^6 e^6+4 a b^5 e \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )\right )}{4 e^7 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.019, size = 669, normalized size = 1.9 \begin{align*}{\frac{-2\,{a}^{6}{e}^{6}+22\,{b}^{6}{d}^{6}-330\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+80\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+160\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-120\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+24\,xa{b}^{5}{d}^{4}{e}^{2}+252\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-20\,{x}^{4}a{b}^{5}d{e}^{5}-160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+60\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+120\,x{a}^{4}{b}^{2}d{e}^{5}-240\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}+360\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+60\,\ln \left ( ex+d \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}-240\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e-240\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}+360\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-240\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+120\,\ln \left ( ex+d \right ) x{a}^{4}{b}^{2}d{e}^{5}-480\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+720\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-480\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}-68\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-24\,x{a}^{5}b{e}^{6}-16\,x{b}^{6}{d}^{5}e+8\,{x}^{5}a{b}^{5}{e}^{6}-2\,{x}^{5}{b}^{6}d{e}^{5}+30\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+5\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+80\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-20\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-12\,d{e}^{5}{a}^{5}b-200\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+210\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-108\,a{b}^{5}{d}^{5}e+90\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+{x}^{6}{b}^{6}{e}^{6}+120\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+60\,\ln \left ( ex+d \right ){x}^{2}{a}^{4}{b}^{2}{e}^{6}+60\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}}{4\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((b*x+a)^2)^(5/2)*(-2*a^6*e^6+22*b^6*d^6-330*x^2*a^2*b^4*d^2*e^4+80*x^3*a*b^5*d^2*e^4+160*x^2*a^3*b^3*d*e^
5-120*x^3*a^2*b^4*d*e^5+24*x*a*b^5*d^4*e^2+252*x^2*a*b^5*d^3*e^3-20*x^4*a*b^5*d*e^5-160*x*a^3*b^3*d^2*e^4+60*x
*a^2*b^4*d^3*e^3+120*x*a^4*b^2*d*e^5-240*ln(e*x+d)*a^3*b^3*d^3*e^3+360*ln(e*x+d)*a^2*b^4*d^4*e^2+60*ln(e*x+d)*
a^4*b^2*d^2*e^4-240*ln(e*x+d)*a*b^5*d^5*e-240*ln(e*x+d)*x^2*a^3*b^3*d*e^5+360*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4-24
0*ln(e*x+d)*x^2*a*b^5*d^3*e^3+120*ln(e*x+d)*x*a^4*b^2*d*e^5-480*ln(e*x+d)*x*a^3*b^3*d^2*e^4+720*ln(e*x+d)*x*a^
2*b^4*d^3*e^3-480*ln(e*x+d)*x*a*b^5*d^4*e^2-68*x^2*b^6*d^4*e^2-24*x*a^5*b*e^6-16*x*b^6*d^5*e+8*x^5*a*b^5*e^6-2
*x^5*b^6*d*e^5+30*x^4*a^2*b^4*e^6+5*x^4*b^6*d^2*e^4+80*x^3*a^3*b^3*e^6-20*x^3*b^6*d^3*e^3-12*d*e^5*a^5*b-200*a
^3*b^3*d^3*e^3+210*a^2*b^4*d^4*e^2-108*a*b^5*d^5*e+90*a^4*b^2*d^2*e^4+60*ln(e*x+d)*b^6*d^6+x^6*b^6*e^6+120*ln(
e*x+d)*x*b^6*d^5*e+60*ln(e*x+d)*x^2*a^4*b^2*e^6+60*ln(e*x+d)*x^2*b^6*d^4*e^2)/(b*x+a)^5/e^7/(e*x+d)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.62313, size = 1110, normalized size = 3.2 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.14616, size = 687, normalized size = 1.98 \begin{align*} 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 )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (b^{6} x^{4} e^{9} \mathrm{sgn}\left (b x + a\right ) - 4 \, b^{6} d x^{3} e^{8} \mathrm{sgn}\left (b x + a\right ) + 12 \, b^{6} d^{2} x^{2} e^{7} \mathrm{sgn}\left (b x + a\right ) - 40 \, b^{6} d^{3} x e^{6} \mathrm{sgn}\left (b x + a\right ) + 8 \, a b^{5} x^{3} e^{9} \mathrm{sgn}\left (b x + a\right ) - 36 \, a b^{5} d x^{2} e^{8} \mathrm{sgn}\left (b x + a\right ) + 144 \, a b^{5} d^{2} x e^{7} \mathrm{sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} x^{2} e^{9} \mathrm{sgn}\left (b x + a\right ) - 180 \, a^{2} b^{4} d x e^{8} \mathrm{sgn}\left (b x + a\right ) + 80 \, a^{3} b^{3} x e^{9} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} + \frac{{\left (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\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))*e^(-7)*log(abs(x*e + d)) + 1/4*(b^6*x^4*e^9*sgn(b*x + a) - 4*b^6*d*x^3*e^8
*sgn(b*x + a) + 12*b^6*d^2*x^2*e^7*sgn(b*x + a) - 40*b^6*d^3*x*e^6*sgn(b*x + a) + 8*a*b^5*x^3*e^9*sgn(b*x + a)
- 36*a*b^5*d*x^2*e^8*sgn(b*x + a) + 144*a*b^5*d^2*x*e^7*sgn(b*x + a) + 30*a^2*b^4*x^2*e^9*sgn(b*x + a) - 180*
a^2*b^4*d*x*e^8*sgn(b*x + a) + 80*a^3*b^3*x*e^9*sgn(b*x + a))*e^(-12) + 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^(-7)/(x*e + d)^2