3.2002 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=344 \[ -\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]

[Out]

-((b^5*(5*b*d - 6*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^6*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^4) + (2*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)^3) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (20*b^3*(b*d - a*e)^3*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*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 = 0.241979, antiderivative size = 344, 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} \[ -\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^5*(5*b*d - 6*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^6*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^4) + (2*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)^3) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (20*b^3*(b*d - a*e)^3*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*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)^5} \, 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)^5} \, 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)^5} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac{b^5 (5 b d-6 a e)}{e^6}+\frac{b^6 x}{e^5}+\frac{(-b d+a e)^6}{e^6 (d+e x)^5}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^4}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^3}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^2}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{b^5 (5 b d-6 a e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{b^6 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac{2 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)^3}-\frac{15 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac{20 b^3 (b d-a e)^3 \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 \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 = 0.163476, size = 318, normalized size = 0.92 \[ -\frac{\sqrt{(a+b x)^2} \left (-5 a^2 b^4 d e^2 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+20 a^3 b^3 e^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+2 a^5 b e^5 (d+4 e x)+a^6 e^6+2 a b^5 e \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-132 d^4 e^2 x^2+32 d^3 e^3 x^3+68 d^2 e^4 x^4-168 d^5 e x-57 d^6+12 d e^5 x^5-2 e^6 x^6\right )\right )}{4 e^7 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(a + b*x)^2]*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + 20*a^3*b^
3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*
e^3*x^3) + 2*a*b^5*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + b
^6*(-57*d^6 - 168*d^5*e*x - 132*d^4*e^2*x^2 + 32*d^3*e^3*x^3 + 68*d^2*e^4*x^4 + 12*d*e^5*x^5 - 2*e^6*x^6) - 60
*b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e*x]))/(4*e^7*(a + b*x)*(d + e*x)^4)

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Maple [B]  time = 0.018, size = 670, normalized size = 2. \begin{align*}{\frac{-{a}^{6}{e}^{6}+57\,{b}^{6}{d}^{6}+540\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-96\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-120\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+240\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-496\,xa{b}^{5}{d}^{4}{e}^{2}-504\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+96\,{x}^{4}a{b}^{5}d{e}^{5}-80\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+440\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-20\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}-120\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+360\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-720\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+240\,\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}-120\,\ln \left ( ex+d \right ){x}^{4}a{b}^{5}d{e}^{5}+240\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}-480\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}+132\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-8\,x{a}^{5}b{e}^{6}+168\,x{b}^{6}{d}^{5}e+24\,{x}^{5}a{b}^{5}{e}^{6}-12\,{x}^{5}{b}^{6}d{e}^{5}-68\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-80\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-32\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-30\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-2\,d{e}^{5}{a}^{5}b-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+125\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-154\,a{b}^{5}{d}^{5}e-5\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+2\,{x}^{6}{b}^{6}{e}^{6}+240\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){x}^{4}{a}^{2}{b}^{4}{e}^{6}+240\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+360\,\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 ) ^{4}} \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)^5,x)

[Out]

1/4*((b*x+a)^2)^(5/2)*(-a^6*e^6+57*b^6*d^6+540*x^2*a^2*b^4*d^2*e^4-96*x^3*a*b^5*d^2*e^4-120*x^2*a^3*b^3*d*e^5+
240*x^3*a^2*b^4*d*e^5-496*x*a*b^5*d^4*e^2-504*x^2*a*b^5*d^3*e^3+96*x^4*a*b^5*d*e^5-80*x*a^3*b^3*d^2*e^4+440*x*
a^2*b^4*d^3*e^3-20*x*a^4*b^2*d*e^5+60*ln(e*x+d)*a^2*b^4*d^4*e^2-120*ln(e*x+d)*a*b^5*d^5*e+360*ln(e*x+d)*x^2*a^
2*b^4*d^2*e^4-720*ln(e*x+d)*x^2*a*b^5*d^3*e^3+240*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^2-12
0*ln(e*x+d)*x^4*a*b^5*d*e^5+240*ln(e*x+d)*x^3*a^2*b^4*d*e^5-480*ln(e*x+d)*x^3*a*b^5*d^2*e^4+132*x^2*b^6*d^4*e^
2-8*x*a^5*b*e^6+168*x*b^6*d^5*e+24*x^5*a*b^5*e^6-12*x^5*b^6*d*e^5-68*x^4*b^6*d^2*e^4-80*x^3*a^3*b^3*e^6-32*x^3
*b^6*d^3*e^3-30*x^2*a^4*b^2*e^6-2*d*e^5*a^5*b-20*a^3*b^3*d^3*e^3+125*a^2*b^4*d^4*e^2-154*a*b^5*d^5*e-5*a^4*b^2
*d^2*e^4+60*ln(e*x+d)*b^6*d^6+2*x^6*b^6*e^6+240*ln(e*x+d)*x^3*b^6*d^3*e^3+60*ln(e*x+d)*x^4*b^6*d^2*e^4+60*ln(e
*x+d)*x^4*a^2*b^4*e^6+240*ln(e*x+d)*x*b^6*d^5*e+360*ln(e*x+d)*x^2*b^6*d^4*e^2)/(b*x+a)^5/e^7/(e*x+d)^4

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.55968, size = 1158, normalized size = 3.37 \begin{align*} \frac{2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \,{\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \,{\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \,{\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \,{\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e
^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6*d*e^5 - 2*a*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16
*(2*b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a*b^5*d^3*e
^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 4*(42*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^
2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*
d^4*e^2 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e^4 + a^2*b^4*d*e^5)*
x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^2 + 4*(b^6*d^5*e - 2*a*b^5*d^4*e^2 + a^2*b^4*d^3*e
^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*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)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.13993, size = 680, normalized size = 1.98 \begin{align*} 15 \,{\left (b^{6} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{5} d e \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{4} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{6} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) - 10 \, b^{6} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + 12 \, a b^{5} x e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} + \frac{{\left (57 \, b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 154 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 125 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 2 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 80 \,{\left (b^{6} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5} \mathrm{sgn}\left (b x + a\right ) - a^{3} b^{3} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 30 \,{\left (7 \, b^{6} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a b^{5} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 18 \, a^{2} b^{4} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm{sgn}\left (b x + a\right ) - a^{4} b^{2} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 4 \,{\left (47 \, b^{6} d^{5} e \mathrm{sgn}\left (b x + a\right ) - 130 \, a b^{5} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 110 \, a^{2} b^{4} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 20 \, 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 ) - 2 \, a^{5} b e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{4 \,{\left (x e + d\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

15*(b^6*d^2*sgn(b*x + a) - 2*a*b^5*d*e*sgn(b*x + a) + a^2*b^4*e^2*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/2
*(b^6*x^2*e^5*sgn(b*x + a) - 10*b^6*d*x*e^4*sgn(b*x + a) + 12*a*b^5*x*e^5*sgn(b*x + a))*e^(-10) + 1/4*(57*b^6*
d^6*sgn(b*x + a) - 154*a*b^5*d^5*e*sgn(b*x + a) + 125*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*
x + a) - 5*a^4*b^2*d^2*e^4*sgn(b*x + a) - 2*a^5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a) + 80*(b^6*d^3*e^3*
sgn(b*x + a) - 3*a*b^5*d^2*e^4*sgn(b*x + a) + 3*a^2*b^4*d*e^5*sgn(b*x + a) - a^3*b^3*e^6*sgn(b*x + a))*x^3 + 3
0*(7*b^6*d^4*e^2*sgn(b*x + a) - 20*a*b^5*d^3*e^3*sgn(b*x + a) + 18*a^2*b^4*d^2*e^4*sgn(b*x + a) - 4*a^3*b^3*d*
e^5*sgn(b*x + a) - a^4*b^2*e^6*sgn(b*x + a))*x^2 + 4*(47*b^6*d^5*e*sgn(b*x + a) - 130*a*b^5*d^4*e^2*sgn(b*x +
a) + 110*a^2*b^4*d^3*e^3*sgn(b*x + a) - 20*a^3*b^3*d^2*e^4*sgn(b*x + a) - 5*a^4*b^2*d*e^5*sgn(b*x + a) - 2*a^5
*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^4