3.2003 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=344 \[ -\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \]

[Out]

(b^6*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])/(5*e^7*(
a + b*x)*(d + e*x)^5) + (3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^4) - (5*b
^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^3) + (10*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)^2) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(
a + b*x)*(d + e*x)) - (6*b^5*(b*d - a*e)*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 = 0.227199, 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} \[ -\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}+\frac{b^6 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]

[Out]

(b^6*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])/(5*e^7*(
a + b*x)*(d + e*x)^5) + (3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^4) - (5*b
^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^3) + (10*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)^2) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(
a + b*x)*(d + e*x)) - (6*b^5*(b*d - a*e)*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)^6} \, 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)^6} \, 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)^6} \, 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^6}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^6}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac{b^6 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}}{5 e^7 (a+b x) (d+e x)^5}+\frac{3 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^4}-\frac{5 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}+\frac{10 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)^2}-\frac{15 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac{6 b^5 (b d-a e) \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 = 0.185556, size = 315, normalized size = 0.92 \[ -\frac{\sqrt{(a+b x)^2} \left (30 a^2 b^4 e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+10 a^3 b^3 e^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a^5 b e^5 (d+5 e x)+2 a^6 e^6-a b^5 d e \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 10*a^3
*b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10
*d*e^3*x^3 + 5*e^4*x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) +
b^6*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) +
60*b^5*(b*d - a*e)*(d + e*x)^5*Log[d + e*x]))/(10*e^7*(a + b*x)*(d + e*x)^5)

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Maple [B]  time = 0.016, size = 603, normalized size = 1.8 \begin{align*}{\frac{-2\,{a}^{6}{e}^{6}-87\,{b}^{6}{d}^{6}-300\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+900\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-100\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-300\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+625\,xa{b}^{5}{d}^{4}{e}^{2}+1100\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+300\,{x}^{4}a{b}^{5}d{e}^{5}-50\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-150\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-25\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+600\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+300\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}+300\,\ln \left ( ex+d \right ){x}^{4}a{b}^{5}d{e}^{5}+600\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-600\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-15\,x{a}^{5}b{e}^{6}-375\,x{b}^{6}{d}^{5}e+50\,{x}^{5}{b}^{6}d{e}^{5}-150\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-50\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-100\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-400\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-50\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-3\,d{e}^{5}{a}^{5}b-10\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-30\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+137\,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}+10\,{x}^{6}{b}^{6}{e}^{6}-600\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}-300\,\ln \left ( ex+d \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}-300\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+60\,\ln \left ( ex+d \right ){x}^{5}a{b}^{5}{e}^{6}-60\,\ln \left ( ex+d \right ){x}^{5}{b}^{6}d{e}^{5}-600\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}}{10\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{5}} \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)^6,x)

[Out]

1/10*((b*x+a)^2)^(5/2)*(-2*a^6*e^6-87*b^6*d^6-300*x^2*a^2*b^4*d^2*e^4+900*x^3*a*b^5*d^2*e^4-100*x^2*a^3*b^3*d*
e^5-300*x^3*a^2*b^4*d*e^5+625*x*a*b^5*d^4*e^2+1100*x^2*a*b^5*d^3*e^3+300*x^4*a*b^5*d*e^5-50*x*a^3*b^3*d^2*e^4-
150*x*a^2*b^4*d^3*e^3-25*x*a^4*b^2*d*e^5+60*ln(e*x+d)*a*b^5*d^5*e+600*ln(e*x+d)*x^2*a*b^5*d^3*e^3+300*ln(e*x+d
)*x*a*b^5*d^4*e^2+300*ln(e*x+d)*x^4*a*b^5*d*e^5+600*ln(e*x+d)*x^3*a*b^5*d^2*e^4-600*x^2*b^6*d^4*e^2-15*x*a^5*b
*e^6-375*x*b^6*d^5*e+50*x^5*b^6*d*e^5-150*x^4*a^2*b^4*e^6-50*x^4*b^6*d^2*e^4-100*x^3*a^3*b^3*e^6-400*x^3*b^6*d
^3*e^3-50*x^2*a^4*b^2*e^6-3*d*e^5*a^5*b-10*a^3*b^3*d^3*e^3-30*a^2*b^4*d^4*e^2+137*a*b^5*d^5*e-5*a^4*b^2*d^2*e^
4-60*ln(e*x+d)*b^6*d^6+10*x^6*b^6*e^6-600*ln(e*x+d)*x^3*b^6*d^3*e^3-300*ln(e*x+d)*x^4*b^6*d^2*e^4-300*ln(e*x+d
)*x*b^6*d^5*e+60*ln(e*x+d)*x^5*a*b^5*e^6-60*ln(e*x+d)*x^5*b^6*d*e^5-600*ln(e*x+d)*x^2*b^6*d^4*e^2)/(b*x+a)^5/e
^7/(e*x+d)^5

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56396, size = 1095, normalized size = 3.18 \begin{align*} \frac{10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \,{\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \,{\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \,{\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - a b^{5} d^{5} e +{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \,{\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/10*(10*b^6*e^6*x^6 + 50*b^6*d*e^5*x^5 - 87*b^6*d^6 + 137*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 10*a^3*b^3*d^3*e
^3 - 5*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - 2*a^6*e^6 - 50*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 - 10
0*(4*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 50*(12*b^6*d^4*e^2 - 22*a*b^5*d^3*e^
3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 - 5*(75*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*a^2*b^4*
d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - a*b^5*d^5*e + (b^6*d*e^5 - a*b
^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - a*b^5*d*e^5)*x^4 + 10*(b^6*d^3*e^3 - a*b^5*d^2*e^4)*x^3 + 10*(b^6*d^4*e^2 - a*b
^5*d^3*e^3)*x^2 + 5*(b^6*d^5*e - a*b^5*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 +
10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.15803, size = 674, normalized size = 1.96 \begin{align*} b^{6} x e^{\left (-6\right )} \mathrm{sgn}\left (b x + a\right ) - 6 \,{\left (b^{6} d \mathrm{sgn}\left (b x + a\right ) - a b^{5} e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (87 \, b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 137 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, 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 ) + 3 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 150 \,{\left (b^{6} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{4} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 9 \, 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} + 50 \,{\left (13 \, b^{6} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 22 \, a b^{5} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 2 \, 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} + 5 \,{\left (77 \, b^{6} d^{5} e \mathrm{sgn}\left (b x + a\right ) - 125 \, a b^{5} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 30 \, 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 ) + 3 \, a^{5} b e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

b^6*x*e^(-6)*sgn(b*x + a) - 6*(b^6*d*sgn(b*x + a) - a*b^5*e*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) - 1/10*(87*
b^6*d^6*sgn(b*x + a) - 137*a*b^5*d^5*e*sgn(b*x + a) + 30*a^2*b^4*d^4*e^2*sgn(b*x + a) + 10*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) + 3*a^5*b*d*e^5*sgn(b*x + a) + 2*a^6*e^6*sgn(b*x + a) + 150*(b^6*d^
2*e^4*sgn(b*x + a) - 2*a*b^5*d*e^5*sgn(b*x + a) + a^2*b^4*e^6*sgn(b*x + a))*x^4 + 100*(5*b^6*d^3*e^3*sgn(b*x +
a) - 9*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 + 50*(13*b^6
*d^4*e^2*sgn(b*x + a) - 22*a*b^5*d^3*e^3*sgn(b*x + a) + 6*a^2*b^4*d^2*e^4*sgn(b*x + a) + 2*a^3*b^3*d*e^5*sgn(b
*x + a) + a^4*b^2*e^6*sgn(b*x + a))*x^2 + 5*(77*b^6*d^5*e*sgn(b*x + a) - 125*a*b^5*d^4*e^2*sgn(b*x + a) + 30*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) + 3*a^5*b*e^6*sgn
(b*x + a))*x)*e^(-7)/(x*e + d)^5