3.1576 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^2} \, dx\)

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

[Out]

(-10*b^2*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + ((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (5*b^3*(b*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a
+ b*x)) - (5*b^4*(b*d - a*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) + (b^5*(d + e*x)^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)) + (5*b*(b*d - a*e)^4*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 = 0.210584, 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} \[ \frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^6 (a+b x)}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^6 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{e^6 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) (d+e x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*b^2*(b*d - a*e)^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + ((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (5*b^3*(b*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a
+ b*x)) - (5*b^4*(b*d - a*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) + (b^5*(d + e*x)^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)) + (5*b*(b*d - a*e)^4*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)^2} \, 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)^2} \, 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{10 b^7 (b d-a e)^3}{e^5}-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^2}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)}+\frac{10 b^8 (b d-a e)^2 (d+e x)}{e^5}-\frac{5 b^9 (b d-a e) (d+e x)^2}{e^5}+\frac{b^{10} (d+e x)^3}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{10 b^2 (b d-a e)^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac{5 b^3 (b d-a e)^2 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{5 b^4 (b d-a e) (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac{b^5 (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x)}+\frac{5 b (b d-a e)^4 \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 = 0.160589, size = 246, normalized size = 0.84 \[ \frac{\sqrt{(a+b x)^2} \left (60 a^2 b^3 e^2 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^4 b d e^4-12 a^5 e^5+20 a b^4 e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(60*a^4*b*d*e^4 - 12*a^5*e^5 + 120*a^3*b^2*e^3*(-d^2 + d*e*x + e^2*x^2) + 60*a^2*b^3*e^2*(2
*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 20*a*b^4*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4
*x^4) + b^5*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + 60*b*(b*d - a*
e)^4*(d + e*x)*Log[d + e*x]))/(12*e^6*(a + b*x)*(d + e*x))

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Maple [B]  time = 0.203, size = 456, normalized size = 1.6 \begin{align*}{\frac{360\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-240\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}-240\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{4}-12\,{a}^{5}{e}^{5}+12\,{b}^{5}{d}^{5}+60\,d{e}^{4}{a}^{4}b-48\,x{b}^{5}{d}^{4}e+20\,{x}^{4}a{b}^{4}{e}^{5}-5\,{x}^{4}{b}^{5}d{e}^{4}+60\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+10\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+120\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-30\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ) x{a}^{4}b{e}^{5}+60\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e+120\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-40\,{x}^{3}a{b}^{4}d{e}^{4}+60\,\ln \left ( ex+d \right ){a}^{4}bd{e}^{4}+120\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-120\,{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}+180\,xa{b}^{4}{d}^{3}{e}^{2}-240\,\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}{e}^{3}+360\,\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{2}-240\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e+120\,x{a}^{3}{b}^{2}d{e}^{4}-240\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-60\,a{b}^{4}{d}^{4}e}{12\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) } \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)^2,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(360*ln(e*x+d)*x*a^2*b^3*d^2*e^3-240*ln(e*x+d)*x*a*b^4*d^3*e^2-240*ln(e*x+d)*x*a^3*b^2*
d*e^4-12*a^5*e^5+12*b^5*d^5+60*d*e^4*a^4*b-48*x*b^5*d^4*e+20*x^4*a*b^4*e^5-5*x^4*b^5*d*e^4+60*x^3*a^2*b^3*e^5+
10*x^3*b^5*d^2*e^3+120*x^2*a^3*b^2*e^5-30*x^2*b^5*d^3*e^2+60*ln(e*x+d)*x*a^4*b*e^5+60*ln(e*x+d)*x*b^5*d^4*e+12
0*x^2*a*b^4*d^2*e^3-180*x^2*a^2*b^3*d*e^4-40*x^3*a*b^4*d*e^4+60*ln(e*x+d)*a^4*b*d*e^4+120*a^2*b^3*d^3*e^2-120*
a^3*b^2*d^2*e^3+60*ln(e*x+d)*b^5*d^5+3*x^5*b^5*e^5+180*x*a*b^4*d^3*e^2-240*ln(e*x+d)*a^3*b^2*d^2*e^3+360*ln(e*
x+d)*a^2*b^3*d^3*e^2-240*ln(e*x+d)*a*b^4*d^4*e+120*x*a^3*b^2*d*e^4-240*x*a^2*b^3*d^2*e^3-60*a*b^4*d^4*e)/(b*x+
a)^5/e^6/(e*x+d)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50243, size = 767, normalized size = 2.63 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \,{\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \,{\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \,{\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \,{\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} +{\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d 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)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*e^5*x^5 + 12*b^5*d^5 - 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4
- 12*a^5*e^5 - 5*(b^5*d*e^4 - 4*a*b^4*e^5)*x^4 + 10*(b^5*d^2*e^3 - 4*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 - 30*(b
^5*d^3*e^2 - 4*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 4*a^3*b^2*e^5)*x^2 - 12*(4*b^5*d^4*e - 15*a*b^4*d^3*e^2 + 20*
a^2*b^3*d^2*e^3 - 10*a^3*b^2*d*e^4)*x + 60*(b^5*d^5 - 4*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 - 4*a^3*b^2*d^2*e^3 +
a^4*b*d*e^4 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))
/(e^7*x + d*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)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.21518, size = 516, normalized size = 1.77 \begin{align*} 5 \,{\left (b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{5} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) - 8 \, b^{5} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 18 \, b^{5} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 48 \, b^{5} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 20 \, a b^{4} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) - 60 \, a b^{4} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 180 \, a b^{4} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 60 \, a^{2} b^{3} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) - 240 \, a^{2} b^{3} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 120 \, a^{3} b^{2} x e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-8\right )} + \frac{{\left (b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 10 \, 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 ) - a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{x e + d} \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)^2,x, algorithm="giac")

[Out]

5*(b^5*d^4*sgn(b*x + a) - 4*a*b^4*d^3*e*sgn(b*x + a) + 6*a^2*b^3*d^2*e^2*sgn(b*x + a) - 4*a^3*b^2*d*e^3*sgn(b*
x + a) + a^4*b*e^4*sgn(b*x + a))*e^(-6)*log(abs(x*e + d)) + 1/12*(3*b^5*x^4*e^6*sgn(b*x + a) - 8*b^5*d*x^3*e^5
*sgn(b*x + a) + 18*b^5*d^2*x^2*e^4*sgn(b*x + a) - 48*b^5*d^3*x*e^3*sgn(b*x + a) + 20*a*b^4*x^3*e^6*sgn(b*x + a
) - 60*a*b^4*d*x^2*e^5*sgn(b*x + a) + 180*a*b^4*d^2*x*e^4*sgn(b*x + a) + 60*a^2*b^3*x^2*e^6*sgn(b*x + a) - 240
*a^2*b^3*d*x*e^5*sgn(b*x + a) + 120*a^3*b^2*x*e^6*sgn(b*x + a))*e^(-8) + (b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e
*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a)
- a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)