3.1580 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=300 \[ \frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5) - (5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^4) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*
(a + b*x)*(d + e*x)^3) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)^2) + (5*
b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (b^5*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 = 0.143804, antiderivative size = 300, 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} \[ \frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5) - (5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^4) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*
(a + b*x)*(d + e*x)^3) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)^2) + (5*
b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)) + (b^5*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)^6} \, 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)^6} \, 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^5 (b d-a e)^5}{e^5 (d+e x)^6}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^5}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^4}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^3}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)^2}+\frac{b^{10}}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}-\frac{5 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}+\frac{10 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}+\frac{5 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}+\frac{b^5 \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 = 0.129879, size = 196, normalized size = 0.65 \[ \frac{\sqrt{(a+b x)^2} \left ((b d-a e) \left (a^2 b^2 e^2 \left (47 d^2+175 d e x+200 e^2 x^2\right )+3 a^3 b e^3 (9 d+25 e x)+12 a^4 e^4+a b^3 e \left (325 d^2 e x+77 d^3+500 d e^2 x^2+300 e^3 x^3\right )+b^4 \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^5 (d+e x)^5 \log (d+e x)\right )}{60 e^6 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*((b*d - a*e)*(12*a^4*e^4 + 3*a^3*b*e^3*(9*d + 25*e*x) + a^2*b^2*e^2*(47*d^2 + 175*d*e*x + 2
00*e^2*x^2) + a*b^3*e*(77*d^3 + 325*d^2*e*x + 500*d*e^2*x^2 + 300*e^3*x^3) + b^4*(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)) + 60*b^5*(d + e*x)^5*Log[d + e*x]))/(60*e^6*(a + b*x)*(d + e*x)^5
)

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Maple [A]  time = 0.2, size = 383, normalized size = 1.3 \begin{align*}{\frac{600\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}-12\,{a}^{5}{e}^{5}+137\,{b}^{5}{d}^{5}-15\,d{e}^{4}{a}^{4}b+60\,\ln \left ( ex+d \right ){x}^{5}{b}^{5}{e}^{5}+625\,x{b}^{5}{d}^{4}e-300\,{x}^{4}a{b}^{4}{e}^{5}+300\,{x}^{4}{b}^{5}d{e}^{4}-300\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+900\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-200\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+1100\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-75\,x{a}^{4}b{e}^{5}+300\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e-600\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-300\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-600\,{x}^{3}a{b}^{4}d{e}^{4}-30\,{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}-300\,xa{b}^{4}{d}^{3}{e}^{2}-100\,x{a}^{3}{b}^{2}d{e}^{4}-150\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+600\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}+300\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}-60\,a{b}^{4}{d}^{4}e}{60\, \left ( bx+a \right ) ^{5}{e}^{6} \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^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(600*ln(e*x+d)*x^2*b^5*d^3*e^2-12*a^5*e^5+137*b^5*d^5-15*d*e^4*a^4*b+60*ln(e*x+d)*x^5*b
^5*e^5+625*x*b^5*d^4*e-300*x^4*a*b^4*e^5+300*x^4*b^5*d*e^4-300*x^3*a^2*b^3*e^5+900*x^3*b^5*d^2*e^3-200*x^2*a^3
*b^2*e^5+1100*x^2*b^5*d^3*e^2-75*x*a^4*b*e^5+300*ln(e*x+d)*x*b^5*d^4*e-600*x^2*a*b^4*d^2*e^3-300*x^2*a^2*b^3*d
*e^4-600*x^3*a*b^4*d*e^4-30*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^5-300*x*a*b^4*d^3*e^2-100*x*
a^3*b^2*d*e^4-150*x*a^2*b^3*d^2*e^3+600*ln(e*x+d)*x^3*b^5*d^2*e^3+300*ln(e*x+d)*x^4*b^5*d*e^4-60*a*b^4*d^4*e)/
(b*x+a)^5/e^6/(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^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 [A]  time = 1.56002, size = 774, normalized size = 2.58 \begin{align*} \frac{137 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (3 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (11 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} - 3 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (25 \, b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 3 \, a^{4} b e^{5}\right )} x + 60 \,{\left (b^{5} e^{5} x^{5} + 5 \, b^{5} d e^{4} x^{4} + 10 \, b^{5} d^{2} e^{3} x^{3} + 10 \, b^{5} d^{3} e^{2} x^{2} + 5 \, b^{5} d^{4} e x + b^{5} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/60*(137*b^5*d^5 - 60*a*b^4*d^4*e - 30*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 15*a^4*b*d*e^4 - 12*a^5*e^5 + 3
00*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(3*b^5*d^2*e^3 - 2*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 100*(11*b^5*d^3*e^2 -
6*a*b^4*d^2*e^3 - 3*a^2*b^3*d*e^4 - 2*a^3*b^2*e^5)*x^2 + 25*(25*b^5*d^4*e - 12*a*b^4*d^3*e^2 - 6*a^2*b^3*d^2*
e^3 - 4*a^3*b^2*d*e^4 - 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + 5*b^5*d*e^4*x^4 + 10*b^5*d^2*e^3*x^3 + 10*b^5*d^3*e
^2*x^2 + 5*b^5*d^4*e*x + b^5*d^5)*log(e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5
*d^4*e^7*x + d^5*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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.15964, size = 510, normalized size = 1.7 \begin{align*} b^{5} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (300 \,{\left (b^{5} d e^{3} \mathrm{sgn}\left (b x + a\right ) - a b^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{4} + 300 \,{\left (3 \, b^{5} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{3} \mathrm{sgn}\left (b x + a\right ) - a^{2} b^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 100 \,{\left (11 \, b^{5} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 6 \, a b^{4} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{2} b^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 2 \, a^{3} b^{2} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 25 \,{\left (25 \, b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) - 12 \, 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 ) - 3 \, a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x +{\left (137 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 60 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 30 \, 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 ) - 15 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) - 12 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

b^5*e^(-6)*log(abs(x*e + d))*sgn(b*x + a) + 1/60*(300*(b^5*d*e^3*sgn(b*x + a) - a*b^4*e^4*sgn(b*x + a))*x^4 +
300*(3*b^5*d^2*e^2*sgn(b*x + a) - 2*a*b^4*d*e^3*sgn(b*x + a) - a^2*b^3*e^4*sgn(b*x + a))*x^3 + 100*(11*b^5*d^3
*e*sgn(b*x + a) - 6*a*b^4*d^2*e^2*sgn(b*x + a) - 3*a^2*b^3*d*e^3*sgn(b*x + a) - 2*a^3*b^2*e^4*sgn(b*x + a))*x^
2 + 25*(25*b^5*d^4*sgn(b*x + a) - 12*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) - 3*a^4*b*e^4*sgn(b*x + a))*x + (137*b^5*d^5*sgn(b*x + a) - 60*a*b^4*d^4*e*sgn(b*x + a) - 30*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) - 15*a^4*b*d*e^4*sgn(b*x + a) - 12*a^5*e^5*sgn(b
*x + a))*e^(-1))*e^(-5)/(x*e + d)^5