3.2001 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=345 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]

[Out]

(15*b^4*(b*d - a*e)^2*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])/(3*e^7*(a + b*x)*(d + e*x)^3) + (3*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)^2) - (15*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*b^5*(b*d - a*e)
*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(3*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*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 = 0.250281, antiderivative size = 345, 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} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^4*(b*d - a*e)^2*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])/(3*e^7*(a + b*x)*(d + e*x)^3) + (3*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)^2) - (15*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*b^5*(b*d - a*e)
*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(3*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*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)^4} \, 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)^4} \, 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)^4} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{15 b^4 (b d-a e)^2}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^4}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac{6 b^5 (b d-a e) (d+e x)}{e^6}+\frac{b^6 (d+e x)^2}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{15 b^4 (b d-a e)^2 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}}{3 e^7 (a+b x) (d+e x)^3}+\frac{3 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)^2}-\frac{15 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)}-\frac{3 b^5 (b d-a e) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{b^6 (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{20 b^3 (b d-a e)^3 \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 = 0.158884, size = 320, normalized size = 0.93 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b^4 e^2 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )+3 a^5 b e^5 (d+3 e x)+a^6 e^6-3 a b^5 e \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+51 d^5 e x+37 d^6+3 d e^5 x^5-e^6 x^6\right )\right )}{3 e^7 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(a + b*x)^2]*(a^6*e^6 + 3*a^5*b*e^5*(d + 3*e*x) + 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) - 10*a^3*b
^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3
- 3*e^4*x^4) - 3*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b
^6*(37*d^6 + 51*d^5*e*x - 39*d^4*e^2*x^2 - 73*d^3*e^3*x^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6) + 60*b^3*(
b*d - a*e)^3*(d + e*x)^3*Log[d + e*x]))/(3*e^7*(a + b*x)*(d + e*x)^3)

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Maple [B]  time = 0.016, size = 692, normalized size = 2. \begin{align*}{\frac{-{a}^{6}{e}^{6}-37\,{b}^{6}{d}^{6}-135\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-189\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+180\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+135\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+243\,xa{b}^{5}{d}^{4}{e}^{2}-27\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-45\,{x}^{4}a{b}^{5}d{e}^{5}+270\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-405\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-45\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}-180\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+180\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+180\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}-540\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+540\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+180\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-540\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+540\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}-180\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+180\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}+39\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-9\,x{a}^{5}b{e}^{6}-51\,x{b}^{6}{d}^{5}e+9\,{x}^{5}a{b}^{5}{e}^{6}-3\,{x}^{5}{b}^{6}d{e}^{5}+45\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+15\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+73\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-45\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-3\,d{e}^{5}{a}^{5}b+110\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-195\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+141\,a{b}^{5}{d}^{5}e-15\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+{x}^{6}{b}^{6}{e}^{6}-60\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}-180\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e-180\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}}{3\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{3}} \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)^4,x)

[Out]

1/3*((b*x+a)^2)^(5/2)*(-a^6*e^6-37*b^6*d^6-135*x^2*a^2*b^4*d^2*e^4-189*x^3*a*b^5*d^2*e^4+180*x^2*a^3*b^3*d*e^5
+135*x^3*a^2*b^4*d*e^5+243*x*a*b^5*d^4*e^2-27*x^2*a*b^5*d^3*e^3-45*x^4*a*b^5*d*e^5+270*x*a^3*b^3*d^2*e^4-405*x
*a^2*b^4*d^3*e^3-45*x*a^4*b^2*d*e^5+60*ln(e*x+d)*a^3*b^3*d^3*e^3-180*ln(e*x+d)*a^2*b^4*d^4*e^2+180*ln(e*x+d)*a
*b^5*d^5*e+180*ln(e*x+d)*x^2*a^3*b^3*d*e^5-540*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4+540*ln(e*x+d)*x^2*a*b^5*d^3*e^3+1
80*ln(e*x+d)*x*a^3*b^3*d^2*e^4-540*ln(e*x+d)*x*a^2*b^4*d^3*e^3+540*ln(e*x+d)*x*a*b^5*d^4*e^2-180*ln(e*x+d)*x^3
*a^2*b^4*d*e^5+180*ln(e*x+d)*x^3*a*b^5*d^2*e^4+39*x^2*b^6*d^4*e^2-9*x*a^5*b*e^6-51*x*b^6*d^5*e+9*x^5*a*b^5*e^6
-3*x^5*b^6*d*e^5+45*x^4*a^2*b^4*e^6+15*x^4*b^6*d^2*e^4+73*x^3*b^6*d^3*e^3-45*x^2*a^4*b^2*e^6-3*d*e^5*a^5*b+110
*a^3*b^3*d^3*e^3-195*a^2*b^4*d^4*e^2+141*a*b^5*d^5*e-15*a^4*b^2*d^2*e^4-60*ln(e*x+d)*b^6*d^6+x^6*b^6*e^6-60*ln
(e*x+d)*x^3*b^6*d^3*e^3+60*ln(e*x+d)*x^3*a^3*b^3*e^6-180*ln(e*x+d)*x*b^6*d^5*e-180*ln(e*x+d)*x^2*b^6*d^4*e^2)/
(b*x+a)^5/e^7/(e*x+d)^3

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.55695, size = 1161, normalized size = 3.37 \begin{align*} \frac{b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} +{\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 110*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e
^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6*d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e
^6)*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*(13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 -
45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^
4*d^3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d
^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^
2 - 3*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*
e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*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)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.15168, size = 679, normalized size = 1.97 \begin{align*} -20 \,{\left (b^{6} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{3} b^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{6} x^{3} e^{8} \mathrm{sgn}\left (b x + a\right ) - 6 \, b^{6} d x^{2} e^{7} \mathrm{sgn}\left (b x + a\right ) + 30 \, b^{6} d^{2} x e^{6} \mathrm{sgn}\left (b x + a\right ) + 9 \, a b^{5} x^{2} e^{8} \mathrm{sgn}\left (b x + a\right ) - 72 \, a b^{5} d x e^{7} \mathrm{sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} x e^{8} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac{{\left (37 \, b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 110 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, 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 ) + a^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (b^{6} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, 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 ) - 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} + 9 \,{\left (9 \, b^{6} d^{5} e \mathrm{sgn}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 30 \, 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 )}}{3 \,{\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

-20*(b^6*d^3*sgn(b*x + a) - 3*a*b^5*d^2*e*sgn(b*x + a) + 3*a^2*b^4*d*e^2*sgn(b*x + a) - a^3*b^3*e^3*sgn(b*x +
a))*e^(-7)*log(abs(x*e + d)) + 1/3*(b^6*x^3*e^8*sgn(b*x + a) - 6*b^6*d*x^2*e^7*sgn(b*x + a) + 30*b^6*d^2*x*e^6
*sgn(b*x + a) + 9*a*b^5*x^2*e^8*sgn(b*x + a) - 72*a*b^5*d*x*e^7*sgn(b*x + a) + 45*a^2*b^4*x*e^8*sgn(b*x + a))*
e^(-12) - 1/3*(37*b^6*d^6*sgn(b*x + a) - 141*a*b^5*d^5*e*sgn(b*x + a) + 195*a^2*b^4*d^4*e^2*sgn(b*x + a) - 110
*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) + 3*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x
+ a) + 45*(b^6*d^4*e^2*sgn(b*x + a) - 4*a*b^5*d^3*e^3*sgn(b*x + a) + 6*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 + 9*(9*b^6*d^5*e*sgn(b*x + a) - 35*a*b^5*d^4*e^2*sgn(b*x
+ a) + 50*a^2*b^4*d^3*e^3*sgn(b*x + a) - 30*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)^3