### 3.1579 $$\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^5} \, dx$$

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

[Out]

(b^5*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])/(4*e^6*(
a + b*x)*(d + e*x)^4) - (5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^3) + (5*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)^2) - (10*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)) - (5*b^4*(b*d - a*e)*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 = 0.148387, 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} $-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^5*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])/(4*e^6*(
a + b*x)*(d + e*x)^4) - (5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^3) + (5*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)^2) - (10*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)) - (5*b^4*(b*d - a*e)*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)^5} \, 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)^5} \, 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^{10}}{e^5}-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^5}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^4}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^3}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^2}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{b^5 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}}{4 e^6 (a+b x) (d+e x)^4}-\frac{5 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}+\frac{5 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)^2}-\frac{10 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)}-\frac{5 b^4 (b d-a e) \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 = 0.128634, size = 243, normalized size = 0.83 $-\frac{\sqrt{(a+b x)^2} \left (30 a^2 b^3 e^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+10 a^3 b^2 e^3 \left (d^2+4 d e x+6 e^2 x^2\right )+5 a^4 b e^4 (d+4 e x)+3 a^5 e^5-5 a b^4 d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+60 b^4 (d+e x)^4 (b d-a e) \log (d+e x)+b^5 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 5*a^4*b*e^4*(d + 4*e*x) + 10*a^3*b^2*e^3*(d^2 + 4*d*e*x + 6*e^2*x^2) + 30*a^2
*b^3*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a*b^4*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e
^3*x^3) + b^5*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + 60*b^4*(
b*d - a*e)*(d + e*x)^4*Log[d + e*x]))/(12*e^6*(a + b*x)*(d + e*x)^4)

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Maple [B]  time = 0.232, size = 458, normalized size = 1.6 \begin{align*}{\frac{240\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}+360\,\ln \left ( ex+d \right ){x}^{2}a{b}^{4}{d}^{2}{e}^{3}-360\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}-3\,{a}^{5}{e}^{5}-77\,{b}^{5}{d}^{5}-5\,d{e}^{4}{a}^{4}b-248\,x{b}^{5}{d}^{4}e+48\,{x}^{4}{b}^{5}d{e}^{4}-120\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-48\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-60\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-252\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-20\,x{a}^{4}b{e}^{5}-240\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e+540\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+240\,{x}^{3}a{b}^{4}d{e}^{4}-30\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+12\,{x}^{5}{b}^{5}{e}^{5}+440\,xa{b}^{4}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e-40\,x{a}^{3}{b}^{2}d{e}^{4}-120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-240\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{4}a{b}^{4}{e}^{5}-60\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}+240\,\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{4}+125\,a{b}^{4}{d}^{4}e}{12\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) ^{4}} \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)^5,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(240*ln(e*x+d)*x*a*b^4*d^3*e^2+360*ln(e*x+d)*x^2*a*b^4*d^2*e^3-360*ln(e*x+d)*x^2*b^5*d^
3*e^2-3*a^5*e^5-77*b^5*d^5-5*d*e^4*a^4*b-248*x*b^5*d^4*e+48*x^4*b^5*d*e^4-120*x^3*a^2*b^3*e^5-48*x^3*b^5*d^2*e
^3-60*x^2*a^3*b^2*e^5-252*x^2*b^5*d^3*e^2-20*x*a^4*b*e^5-240*ln(e*x+d)*x*b^5*d^4*e+540*x^2*a*b^4*d^2*e^3-180*x
^2*a^2*b^3*d*e^4+240*x^3*a*b^4*d*e^4-30*a^2*b^3*d^3*e^2-10*a^3*b^2*d^2*e^3-60*ln(e*x+d)*b^5*d^5+12*x^5*b^5*e^5
+440*x*a*b^4*d^3*e^2+60*ln(e*x+d)*a*b^4*d^4*e-40*x*a^3*b^2*d*e^4-120*x*a^2*b^3*d^2*e^3-240*ln(e*x+d)*x^3*b^5*d
^2*e^3+60*ln(e*x+d)*x^4*a*b^4*e^5-60*ln(e*x+d)*x^4*b^5*d*e^4+240*ln(e*x+d)*x^3*a*b^4*d*e^4+125*a*b^4*d^4*e)/(b
*x+a)^5/e^6/(e*x+d)^4

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62253, size = 838, normalized size = 2.87 \begin{align*} \frac{12 \, b^{5} e^{5} x^{5} + 48 \, b^{5} d e^{4} x^{4} - 77 \, b^{5} d^{5} + 125 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 24 \,{\left (2 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 5 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \,{\left (21 \, b^{5} d^{3} e^{2} - 45 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \,{\left (62 \, b^{5} d^{4} e - 110 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} d^{5} - a b^{4} d^{4} e +{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \,{\left (b^{5} d^{2} e^{3} - a b^{4} d e^{4}\right )} x^{3} + 6 \,{\left (b^{5} d^{3} e^{2} - a b^{4} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{5} d^{4} e - a b^{4} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(12*b^5*e^5*x^5 + 48*b^5*d*e^4*x^4 - 77*b^5*d^5 + 125*a*b^4*d^4*e - 30*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e
^3 - 5*a^4*b*d*e^4 - 3*a^5*e^5 - 24*(2*b^5*d^2*e^3 - 10*a*b^4*d*e^4 + 5*a^2*b^3*e^5)*x^3 - 12*(21*b^5*d^3*e^2
- 45*a*b^4*d^2*e^3 + 15*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 - 4*(62*b^5*d^4*e - 110*a*b^4*d^3*e^2 + 30*a^2*b^3*
d^2*e^3 + 10*a^3*b^2*d*e^4 + 5*a^4*b*e^5)*x - 60*(b^5*d^5 - a*b^4*d^4*e + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(b^5
*d^2*e^3 - a*b^4*d*e^4)*x^3 + 6*(b^5*d^3*e^2 - a*b^4*d^2*e^3)*x^2 + 4*(b^5*d^4*e - a*b^4*d^3*e^2)*x)*log(e*x +
d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*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)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.21613, size = 500, normalized size = 1.71 \begin{align*} b^{5} x e^{\left (-5\right )} \mathrm{sgn}\left (b x + a\right ) - 5 \,{\left (b^{5} d \mathrm{sgn}\left (b x + a\right ) - a b^{4} e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (77 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 125 \, 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 ) + 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 ) + 3 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 120 \,{\left (b^{5} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{3} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 60 \,{\left (5 \, b^{5} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 9 \, 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} + 20 \,{\left (13 \, b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 22 \, a b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, 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 )}}{12 \,{\left (x e + d\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

b^5*x*e^(-5)*sgn(b*x + a) - 5*(b^5*d*sgn(b*x + a) - a*b^4*e*sgn(b*x + a))*e^(-6)*log(abs(x*e + d)) - 1/12*(77*
b^5*d^5*sgn(b*x + a) - 125*a*b^4*d^4*e*sgn(b*x + a) + 30*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) + 3*a^5*e^5*sgn(b*x + a) + 120*(b^5*d^2*e^3*sgn(b*x + a) - 2*a*b^4*d*e^
4*sgn(b*x + a) + a^2*b^3*e^5*sgn(b*x + a))*x^3 + 60*(5*b^5*d^3*e^2*sgn(b*x + a) - 9*a*b^4*d^2*e^3*sgn(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 + 20*(13*b^5*d^4*e*sgn(b*x + a) - 22*a*b^4*d^3
*e^2*sgn(b*x + a) + 6*a^2*b^3*d^2*e^3*sgn(b*x + a) + 2*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)^4