3.1567 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=200 \[ -\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7} \]

[Out]

((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^7) - (b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^6) + (3*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a +
b*x)*(d + e*x)^5) - (b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^4*(a + b*x)*(d + e*x)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0885224, antiderivative size = 200, 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} \[ -\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^7) - (b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^6) + (3*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a +
b*x)*(d + e*x)^5) - (b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^4*(a + b*x)*(d + e*x)^4)

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

Mathematica [A]  time = 0.0413052, size = 112, normalized size = 0.56 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 b e^2 (d+7 e x)+20 a^3 e^3+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )}{140 e^4 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(20*a^3*e^3 + 10*a^2*b*e^2*(d + 7*e*x) + 4*a*b^2*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + b^3*(d^3
+ 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)))/(140*e^4*(a + b*x)*(d + e*x)^7)

________________________________________________________________________________________

Maple [A]  time = 0.155, size = 131, normalized size = 0.7 \begin{align*} -{\frac{35\,{x}^{3}{b}^{3}{e}^{3}+84\,{x}^{2}a{b}^{2}{e}^{3}+21\,{x}^{2}{b}^{3}d{e}^{2}+70\,x{a}^{2}b{e}^{3}+28\,xa{b}^{2}d{e}^{2}+7\,x{b}^{3}{d}^{2}e+20\,{a}^{3}{e}^{3}+10\,d{e}^{2}{a}^{2}b+4\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{140\,{e}^{4} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/(e*x+d)^8,x)

[Out]

-1/140/e^4*(35*b^3*e^3*x^3+84*a*b^2*e^3*x^2+21*b^3*d*e^2*x^2+70*a^2*b*e^3*x+28*a*b^2*d*e^2*x+7*b^3*d^2*e*x+20*
a^3*e^3+10*a^2*b*d*e^2+4*a*b^2*d^2*e+b^3*d^3)*((b*x+a)^2)^(3/2)/(e*x+d)^7/(b*x+a)^3

________________________________________________________________________________________

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)^(3/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.85414, size = 382, normalized size = 1.91 \begin{align*} -\frac{35 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 4 \, a b^{2} d^{2} e + 10 \, a^{2} b d e^{2} + 20 \, a^{3} e^{3} + 21 \,{\left (b^{3} d e^{2} + 4 \, a b^{2} e^{3}\right )} x^{2} + 7 \,{\left (b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 10 \, a^{2} b e^{3}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\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)^(3/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*e^3*x^3 + b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2 + 20*a^3*e^3 + 21*(b^3*d*e^2 + 4*a*b^2*e^3)*
x^2 + 7*(b^3*d^2*e + 4*a*b^2*d*e^2 + 10*a^2*b*e^3)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*x^5 + 35*d^3*e^8*x
^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

________________________________________________________________________________________

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)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18464, size = 228, normalized size = 1.14 \begin{align*} -\frac{{\left (35 \, b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 7 \, b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) + 84 \, a b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, a b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{2} b x e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{140 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^3*e^3*sgn(b*x + a) + 21*b^3*d*x^2*e^2*sgn(b*x + a) + 7*b^3*d^2*x*e*sgn(b*x + a) + b^3*d^3*sgn
(b*x + a) + 84*a*b^2*x^2*e^3*sgn(b*x + a) + 28*a*b^2*d*x*e^2*sgn(b*x + a) + 4*a*b^2*d^2*e*sgn(b*x + a) + 70*a^
2*b*x*e^3*sgn(b*x + a) + 10*a^2*b*d*e^2*sgn(b*x + a) + 20*a^3*e^3*sgn(b*x + a))*e^(-4)/(x*e + d)^7