3.1578 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=292 \[ -\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (4 b d-5 a e)}{e^5 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^3}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)} \]

[Out]

-((b^4*(4*b*d - 5*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x))) + (b^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^4*(a + b*x)) + ((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^3) - (5*b*(b
*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^2) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (10*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x]
)/(e^6*(a + b*x))

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Rubi [A]  time = 0.157445, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ -\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (4 b d-5 a e)}{e^5 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^3}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^4*(4*b*d - 5*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x))) + (b^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^4*(a + b*x)) + ((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^3) - (5*b*(b
*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^2) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (10*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x]
)/(e^6*(a + b*x))

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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{\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{\left (a b+b^2 x\right )^5}{(d+e x)^4} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^9 (4 b d-5 a e)}{e^5}+\frac{b^{10} x}{e^4}-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^4}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^3}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^2}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{b^4 (4 b d-5 a e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{b^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)}+\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^2}+\frac{10 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac{10 b^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.118378, size = 247, normalized size = 0.85 \[ \frac{\sqrt{(a+b x)^2} \left (10 a^2 b^3 d e^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )-20 a^3 b^2 e^3 \left (d^2+3 d e x+3 e^2 x^2\right )-5 a^4 b e^4 (d+3 e x)-2 a^5 e^5+10 a b^4 e \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )+60 b^3 (d+e x)^3 (b d-a e)^2 \log (d+e x)+b^5 \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(-2*a^5*e^5 - 5*a^4*b*e^4*(d + 3*e*x) - 20*a^3*b^2*e^3*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^2
*b^3*d*e^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 10*a*b^4*e*(-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) + b^5*(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) + 60*b^3*(
b*d - a*e)^2*(d + e*x)^3*Log[d + e*x]))/(6*e^6*(a + b*x)*(d + e*x)^3)

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Maple [B]  time = 0.2, size = 502, normalized size = 1.7 \begin{align*}{\frac{180\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-360\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}+180\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{4}-360\,\ln \left ( ex+d \right ){x}^{2}a{b}^{4}{d}^{2}{e}^{3}+180\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}-2\,{a}^{5}{e}^{5}+47\,{b}^{5}{d}^{5}-5\,d{e}^{4}{a}^{4}b+81\,x{b}^{5}{d}^{4}e+30\,{x}^{4}a{b}^{4}{e}^{5}-15\,{x}^{4}{b}^{5}d{e}^{4}-63\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-60\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-9\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-15\,x{a}^{4}b{e}^{5}+180\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e-90\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+90\,{x}^{3}a{b}^{4}d{e}^{4}+110\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-20\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+3\,{x}^{5}{b}^{5}{e}^{5}-270\,xa{b}^{4}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{2}-120\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e-60\,x{a}^{3}{b}^{2}d{e}^{4}+270\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{3}{e}^{5}+60\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}-120\,\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{4}-130\,a{b}^{4}{d}^{4}e}{6\, \left ( bx+a \right ) ^{5}{e}^{6} \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^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x)

[Out]

1/6*((b*x+a)^2)^(5/2)*(180*ln(e*x+d)*x*a^2*b^3*d^2*e^3-360*ln(e*x+d)*x*a*b^4*d^3*e^2+180*ln(e*x+d)*x^2*a^2*b^3
*d*e^4-360*ln(e*x+d)*x^2*a*b^4*d^2*e^3+180*ln(e*x+d)*x^2*b^5*d^3*e^2-2*a^5*e^5+47*b^5*d^5-5*d*e^4*a^4*b+81*x*b
^5*d^4*e+30*x^4*a*b^4*e^5-15*x^4*b^5*d*e^4-63*x^3*b^5*d^2*e^3-60*x^2*a^3*b^2*e^5-9*x^2*b^5*d^3*e^2-15*x*a^4*b*
e^5+180*ln(e*x+d)*x*b^5*d^4*e-90*x^2*a*b^4*d^2*e^3+180*x^2*a^2*b^3*d*e^4+90*x^3*a*b^4*d*e^4+110*a^2*b^3*d^3*e^
2-20*a^3*b^2*d^2*e^3+60*ln(e*x+d)*b^5*d^5+3*x^5*b^5*e^5-270*x*a*b^4*d^3*e^2+60*ln(e*x+d)*a^2*b^3*d^3*e^2-120*l
n(e*x+d)*a*b^4*d^4*e-60*x*a^3*b^2*d*e^4+270*x*a^2*b^3*d^2*e^3+60*ln(e*x+d)*x^3*a^2*b^3*e^5+60*ln(e*x+d)*x^3*b^
5*d^2*e^3-120*ln(e*x+d)*x^3*a*b^4*d*e^4-130*a*b^4*d^4*e)/(b*x+a)^5/e^6/(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^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 [A]  time = 1.61037, size = 867, normalized size = 2.97 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} + 47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} - 15 \,{\left (b^{5} d e^{4} - 2 \, a b^{4} e^{5}\right )} x^{4} - 9 \,{\left (7 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, b^{5} d^{3} e^{2} + 30 \, a b^{4} d^{2} e^{3} - 60 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (27 \, b^{5} d^{4} e - 90 \, a b^{4} d^{3} e^{2} + 90 \, a^{2} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x + 60 \,{\left (b^{5} d^{5} - 2 \, a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} +{\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (b^{5} d^{3} e^{2} - 2 \, a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4}\right )} x^{2} + 3 \,{\left (b^{5} d^{4} e - 2 \, a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*e^5*x^5 + 47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 -
2*a^5*e^5 - 15*(b^5*d*e^4 - 2*a*b^4*e^5)*x^4 - 9*(7*b^5*d^2*e^3 - 10*a*b^4*d*e^4)*x^3 - 3*(3*b^5*d^3*e^2 + 30
*a*b^4*d^2*e^3 - 60*a^2*b^3*d*e^4 + 20*a^3*b^2*e^5)*x^2 + 3*(27*b^5*d^4*e - 90*a*b^4*d^3*e^2 + 90*a^2*b^3*d^2*
e^3 - 20*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x + 60*(b^5*d^5 - 2*a*b^4*d^4*e + a^2*b^3*d^3*e^2 + (b^5*d^2*e^3 - 2*a*b
^4*d*e^4 + a^2*b^3*e^5)*x^3 + 3*(b^5*d^3*e^2 - 2*a*b^4*d^2*e^3 + a^2*b^3*d*e^4)*x^2 + 3*(b^5*d^4*e - 2*a*b^4*d
^3*e^2 + a^2*b^3*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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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**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 = 1.25927, size = 506, normalized size = 1.73 \begin{align*} 10 \,{\left (b^{5} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{4} d e \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{3} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{5} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 8 \, b^{5} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a b^{4} x e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-8\right )} + \frac{{\left (47 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 130 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 110 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 5 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) - 2 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 60 \,{\left (b^{5} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{4} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{4} \mathrm{sgn}\left (b x + a\right ) - a^{3} b^{2} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 15 \,{\left (7 \, b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 20 \, a b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \, a^{2} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{4} \mathrm{sgn}\left (b x + a\right ) - a^{4} b e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

10*(b^5*d^2*sgn(b*x + a) - 2*a*b^4*d*e*sgn(b*x + a) + a^2*b^3*e^2*sgn(b*x + a))*e^(-6)*log(abs(x*e + d)) + 1/2
*(b^5*x^2*e^4*sgn(b*x + a) - 8*b^5*d*x*e^3*sgn(b*x + a) + 10*a*b^4*x*e^4*sgn(b*x + a))*e^(-8) + 1/6*(47*b^5*d^
5*sgn(b*x + a) - 130*a*b^4*d^4*e*sgn(b*x + a) + 110*a^2*b^3*d^3*e^2*sgn(b*x + a) - 20*a^3*b^2*d^2*e^3*sgn(b*x
+ a) - 5*a^4*b*d*e^4*sgn(b*x + a) - 2*a^5*e^5*sgn(b*x + a) + 60*(b^5*d^3*e^2*sgn(b*x + a) - 3*a*b^4*d^2*e^3*sg
n(b*x + a) + 3*a^2*b^3*d*e^4*sgn(b*x + a) - a^3*b^2*e^5*sgn(b*x + a))*x^2 + 15*(7*b^5*d^4*e*sgn(b*x + a) - 20*
a*b^4*d^3*e^2*sgn(b*x + a) + 18*a^2*b^3*d^2*e^3*sgn(b*x + a) - 4*a^3*b^2*d*e^4*sgn(b*x + a) - a^4*b*e^5*sgn(b*
x + a))*x)*e^(-6)/(x*e + d)^3