### 3.1581 $$\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^7} \, dx$$

Optimal. Leaf size=48 $\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)}$

[Out]

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

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Rubi [A]  time = 0.0194524, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {646, 37} $\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [B]  time = 0.0784895, size = 218, normalized size = 4.54 $-\frac{\sqrt{(a+b x)^2} \left (a^2 b^3 e^2 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^4 b e^4 (d+6 e x)+a^5 e^5+a b^4 e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5*e^5 + a^4*b*e^4*(d + 6*e*x) + a^3*b^2*e^3*(d^2 + 6*d*e*x + 15*e^2*x^2) + a^2*b^3*e^2*
(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + a*b^4*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*
e^4*x^4) + b^5*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)))/(6*e^6*(a + b*
x)*(d + e*x)^6)

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Maple [B]  time = 0.155, size = 283, normalized size = 5.9 \begin{align*} -{\frac{6\,{x}^{5}{b}^{5}{e}^{5}+15\,{x}^{4}a{b}^{4}{e}^{5}+15\,{x}^{4}{b}^{5}d{e}^{4}+20\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+20\,{x}^{3}a{b}^{4}d{e}^{4}+20\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+15\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+15\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+15\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+15\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+6\,x{a}^{4}b{e}^{5}+6\,x{a}^{3}{b}^{2}d{e}^{4}+6\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+6\,xa{b}^{4}{d}^{3}{e}^{2}+6\,x{b}^{5}{d}^{4}e+{a}^{5}{e}^{5}+d{e}^{4}{a}^{4}b+{a}^{3}{b}^{2}{d}^{2}{e}^{3}+{a}^{2}{b}^{3}{d}^{3}{e}^{2}+a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{6\, \left ( ex+d \right ) ^{6}{e}^{6} \left ( bx+a \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)^7,x)

[Out]

-1/6*(6*b^5*e^5*x^5+15*a*b^4*e^5*x^4+15*b^5*d*e^4*x^4+20*a^2*b^3*e^5*x^3+20*a*b^4*d*e^4*x^3+20*b^5*d^2*e^3*x^3
+15*a^3*b^2*e^5*x^2+15*a^2*b^3*d*e^4*x^2+15*a*b^4*d^2*e^3*x^2+15*b^5*d^3*e^2*x^2+6*a^4*b*e^5*x+6*a^3*b^2*d*e^4
*x+6*a^2*b^3*d^2*e^3*x+6*a*b^4*d^3*e^2*x+6*b^5*d^4*e*x+a^5*e^5+a^4*b*d*e^4+a^3*b^2*d^2*e^3+a^2*b^3*d^3*e^2+a*b
^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^6/e^6/(b*x+a)^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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64554, size = 595, normalized size = 12.4 \begin{align*} -\frac{6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \,{\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*e^5*x^5 + b^5*d^5 + a*b^4*d^4*e + a^2*b^3*d^3*e^2 + a^3*b^2*d^2*e^3 + a^4*b*d*e^4 + a^5*e^5 + 15*(
b^5*d*e^4 + a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 + a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 15*(b^5*d^3*e^2 + a*b^4*d^2*e^
3 + a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + 6*(b^5*d^4*e + a*b^4*d^3*e^2 + a^2*b^3*d^2*e^3 + a^3*b^2*d*e^4 + a^4*b*
e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*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)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.21132, size = 508, normalized size = 10.58 \begin{align*} -\frac{{\left (6 \, b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 20 \, a b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{4} b x e^{5} \mathrm{sgn}\left (b x + a\right ) + 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 )}}{6 \,{\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5*e^5*sgn(b*x + a) + 15*b^5*d*x^4*e^4*sgn(b*x + a) + 20*b^5*d^2*x^3*e^3*sgn(b*x + a) + 15*b^5*d^
3*x^2*e^2*sgn(b*x + a) + 6*b^5*d^4*x*e*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 15*a*b^4*x^4*e^5*sgn(b*x + a) + 2
0*a*b^4*d*x^3*e^4*sgn(b*x + a) + 15*a*b^4*d^2*x^2*e^3*sgn(b*x + a) + 6*a*b^4*d^3*x*e^2*sgn(b*x + a) + a*b^4*d^
4*e*sgn(b*x + a) + 20*a^2*b^3*x^3*e^5*sgn(b*x + a) + 15*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 6*a^2*b^3*d^2*x*e^3*s
gn(b*x + a) + a^2*b^3*d^3*e^2*sgn(b*x + a) + 15*a^3*b^2*x^2*e^5*sgn(b*x + a) + 6*a^3*b^2*d*x*e^4*sgn(b*x + a)
+ a^3*b^2*d^2*e^3*sgn(b*x + a) + 6*a^4*b*x*e^5*sgn(b*x + a) + a^4*b*d*e^4*sgn(b*x + a) + a^5*e^5*sgn(b*x + a))
*e^(-6)/(x*e + d)^6