3.1998 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=298 \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6 \log (d+e x)}{e^7 (a+b x)}+\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e} \]

[Out]

-((b*(b*d - a*e)^5*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + ((b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4) + ((b*d - a*e)^2*
(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - ((b*d - a*e)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
5*e^2) + ((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e) + ((b*d - a*e)^6*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 = 0.163153, antiderivative size = 298, 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} \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6 \log (d+e x)}{e^7 (a+b x)}+\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(b*d - a*e)^5*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + ((b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4) + ((b*d - a*e)^2*
(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - ((b*d - a*e)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
5*e^2) + ((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e) + ((b*d - a*e)^6*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} \, 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} \, 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} \, 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 (b d-a e)^5}{e^6}+\frac{b (b d-a e)^4 (a+b x)}{e^5}-\frac{b (b d-a e)^3 (a+b x)^2}{e^4}+\frac{b (b d-a e)^2 (a+b x)^3}{e^3}-\frac{b (b d-a e) (a+b x)^4}{e^2}+\frac{b (a+b x)^5}{e}+\frac{(-b d+a e)^6}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{b (b d-a e)^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{(b d-a e)^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5}-\frac{(b d-a e)^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4}+\frac{(b d-a e)^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3}-\frac{(b d-a e) (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^2}+\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e}+\frac{(b d-a e)^6 \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 = 0.121273, size = 248, normalized size = 0.83 \[ \frac{\sqrt{(a+b x)^2} \left (b e x \left (75 a^2 b^3 e^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+450 a^4 b e^4 (e x-2 d)+360 a^5 e^5+6 a b^4 e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)\right )}{60 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200*a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2
*x^2) + 75*a^2*b^3*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d
^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e
^4*x^4 + 10*e^5*x^5)) + 60*(b*d - a*e)^6*Log[d + e*x]))/(60*e^7*(a + b*x))

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Maple [B]  time = 0.01, size = 428, normalized size = 1.4 \begin{align*}{\frac{450\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+120\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-600\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-300\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+360\,xa{b}^{5}{d}^{4}{e}^{2}-180\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-90\,{x}^{4}a{b}^{5}d{e}^{5}+1200\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-900\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-900\,x{a}^{4}{b}^{2}d{e}^{5}-1200\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}+900\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+900\,\ln \left ( ex+d \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}-360\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e-360\,\ln \left ( ex+d \right ){a}^{5}bd{e}^{5}+30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+360\,x{a}^{5}b{e}^{6}-60\,x{b}^{6}{d}^{5}e+72\,{x}^{5}a{b}^{5}{e}^{6}-12\,{x}^{5}{b}^{6}d{e}^{5}+225\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+15\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-20\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+450\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+10\,{x}^{6}{b}^{6}{e}^{6}+60\,\ln \left ( ex+d \right ){a}^{6}{e}^{6}}{60\, \left ( bx+a \right ) ^{5}{e}^{7}} \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),x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(450*x^2*a^2*b^4*d^2*e^4+120*x^3*a*b^5*d^2*e^4-600*x^2*a^3*b^3*d*e^5-300*x^3*a^2*b^4*d*
e^5+360*x*a*b^5*d^4*e^2-180*x^2*a*b^5*d^3*e^3-90*x^4*a*b^5*d*e^5+1200*x*a^3*b^3*d^2*e^4-900*x*a^2*b^4*d^3*e^3-
900*x*a^4*b^2*d*e^5-1200*ln(e*x+d)*a^3*b^3*d^3*e^3+900*ln(e*x+d)*a^2*b^4*d^4*e^2+900*ln(e*x+d)*a^4*b^2*d^2*e^4
-360*ln(e*x+d)*a*b^5*d^5*e-360*ln(e*x+d)*a^5*b*d*e^5+30*x^2*b^6*d^4*e^2+360*x*a^5*b*e^6-60*x*b^6*d^5*e+72*x^5*
a*b^5*e^6-12*x^5*b^6*d*e^5+225*x^4*a^2*b^4*e^6+15*x^4*b^6*d^2*e^4+400*x^3*a^3*b^3*e^6-20*x^3*b^6*d^3*e^3+450*x
^2*a^4*b^2*e^6+60*ln(e*x+d)*b^6*d^6+10*x^6*b^6*e^6+60*ln(e*x+d)*a^6*e^6)/(b*x+a)^5/e^7

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50691, size = 726, normalized size = 2.44 \begin{align*} \frac{10 \, b^{6} e^{6} x^{6} - 12 \,{\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \,{\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \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),x, algorithm="fricas")

[Out]

1/60*(10*b^6*e^6*x^6 - 12*(b^6*d*e^5 - 6*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^
4 - 20*(b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 - 20*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2 - 6*a*b^5*d^3
*e^3 + 15*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 - 60*(b^6*d^5*e - 6*a*b^5*d^4*e^2 + 15*a^2*
b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^
4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*log(e*x + d))/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),x)

[Out]

Timed out

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Giac [B]  time = 1.16673, size = 705, normalized size = 2.37 \begin{align*}{\left (b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 15 \, 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 ) + 15 \, 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 )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, b^{6} x^{6} e^{5} \mathrm{sgn}\left (b x + a\right ) - 12 \, b^{6} d x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{6} d^{2} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 30 \, b^{6} d^{4} x^{2} e \mathrm{sgn}\left (b x + a\right ) - 60 \, b^{6} d^{5} x \mathrm{sgn}\left (b x + a\right ) + 72 \, a b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) - 90 \, a b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 120 \, a b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 180 \, a b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 360 \, a b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + 225 \, a^{2} b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) - 300 \, a^{2} b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 900 \, a^{2} b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 400 \, a^{3} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) - 600 \, a^{3} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 1200 \, a^{3} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 450 \, a^{4} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) - 900 \, a^{4} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + 360 \, a^{5} b x e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\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),x, algorithm="giac")

[Out]

(b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*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) + 15*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))*e^(-7)*log(abs
(x*e + d)) + 1/60*(10*b^6*x^6*e^5*sgn(b*x + a) - 12*b^6*d*x^5*e^4*sgn(b*x + a) + 15*b^6*d^2*x^4*e^3*sgn(b*x +
a) - 20*b^6*d^3*x^3*e^2*sgn(b*x + a) + 30*b^6*d^4*x^2*e*sgn(b*x + a) - 60*b^6*d^5*x*sgn(b*x + a) + 72*a*b^5*x^
5*e^5*sgn(b*x + a) - 90*a*b^5*d*x^4*e^4*sgn(b*x + a) + 120*a*b^5*d^2*x^3*e^3*sgn(b*x + a) - 180*a*b^5*d^3*x^2*
e^2*sgn(b*x + a) + 360*a*b^5*d^4*x*e*sgn(b*x + a) + 225*a^2*b^4*x^4*e^5*sgn(b*x + a) - 300*a^2*b^4*d*x^3*e^4*s
gn(b*x + a) + 450*a^2*b^4*d^2*x^2*e^3*sgn(b*x + a) - 900*a^2*b^4*d^3*x*e^2*sgn(b*x + a) + 400*a^3*b^3*x^3*e^5*
sgn(b*x + a) - 600*a^3*b^3*d*x^2*e^4*sgn(b*x + a) + 1200*a^3*b^3*d^2*x*e^3*sgn(b*x + a) + 450*a^4*b^2*x^2*e^5*
sgn(b*x + a) - 900*a^4*b^2*d*x*e^4*sgn(b*x + a) + 360*a^5*b*x*e^5*sgn(b*x + a))*e^(-6)