3.2006 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{56 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{8 (d+e x)^8 (b d-a e)} \]

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(56*(b*d - a*e)^2*(d + e*x)^7)

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Rubi [A]  time = 0.0526529, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{56 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{8 (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(56*(b*d - a*e)^2*(d + e*x)^7)

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^9} \, dx}{a b+b^2 x}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{\left (b^2 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{8 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{56 (b d-a e)^2 (d+e x)^7}\\ \end{align*}

Mathematica [B]  time = 0.117155, size = 295, normalized size = 3.01 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 b^4 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+4 a^3 b^3 e^3 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a^5 b e^5 (d+8 e x)+7 a^6 e^6+2 a b^5 e \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )\right )}{56 e^7 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a^3*
b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a^2*b^4*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d
*e^3*x^3 + 70*e^4*x^4) + 2*a*b^5*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*
x^5) + b^6*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)))/
(56*e^7*(a + b*x)*(d + e*x)^8)

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Maple [B]  time = 0.007, size = 392, normalized size = 4. \begin{align*} -{\frac{28\,{x}^{6}{b}^{6}{e}^{6}+112\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}+210\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+140\,{x}^{4}a{b}^{5}d{e}^{5}+70\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+224\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+168\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+112\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+56\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+140\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+112\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+84\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+56\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+28\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+48\,x{a}^{5}b{e}^{6}+40\,x{a}^{4}{b}^{2}d{e}^{5}+32\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+24\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+16\,xa{b}^{5}{d}^{4}{e}^{2}+8\,x{b}^{6}{d}^{5}e+7\,{a}^{6}{e}^{6}+6\,d{e}^{5}{a}^{5}b+5\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+4\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+3\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+2\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{56\,{e}^{7} \left ( ex+d \right ) ^{8} \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*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x)

[Out]

-1/56/e^7*(28*b^6*e^6*x^6+112*a*b^5*e^6*x^5+56*b^6*d*e^5*x^5+210*a^2*b^4*e^6*x^4+140*a*b^5*d*e^5*x^4+70*b^6*d^
2*e^4*x^4+224*a^3*b^3*e^6*x^3+168*a^2*b^4*d*e^5*x^3+112*a*b^5*d^2*e^4*x^3+56*b^6*d^3*e^3*x^3+140*a^4*b^2*e^6*x
^2+112*a^3*b^3*d*e^5*x^2+84*a^2*b^4*d^2*e^4*x^2+56*a*b^5*d^3*e^3*x^2+28*b^6*d^4*e^2*x^2+48*a^5*b*e^6*x+40*a^4*
b^2*d*e^5*x+32*a^3*b^3*d^2*e^4*x+24*a^2*b^4*d^3*e^3*x+16*a*b^5*d^4*e^2*x+8*b^6*d^5*e*x+7*a^6*e^6+6*a^5*b*d*e^5
+5*a^4*b^2*d^2*e^4+4*a^3*b^3*d^3*e^3+3*a^2*b^4*d^4*e^2+2*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^8/(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*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58932, size = 871, normalized size = 8.89 \begin{align*} -\frac{28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \,{\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \,{\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \,{\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \,{\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 +
6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6
)*x^4 + 56*(b^6*d^3*e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6)*x^3 + 28*(b^6*d^4*e^2 + 2*a*b^5*d
^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 8*(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4
*d^3*e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 +
56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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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*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

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Giac [B]  time = 1.13487, size = 702, normalized size = 7.16 \begin{align*} -\frac{{\left (28 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 56 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 70 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 112 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 140 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 112 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 16 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 210 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 168 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 24 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 224 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 112 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 32 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 140 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 40 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 48 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 7 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{56 \,{\left (x e + d\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/56*(28*b^6*x^6*e^6*sgn(b*x + a) + 56*b^6*d*x^5*e^5*sgn(b*x + a) + 70*b^6*d^2*x^4*e^4*sgn(b*x + a) + 56*b^6*
d^3*x^3*e^3*sgn(b*x + a) + 28*b^6*d^4*x^2*e^2*sgn(b*x + a) + 8*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*x + a)
+ 112*a*b^5*x^5*e^6*sgn(b*x + a) + 140*a*b^5*d*x^4*e^5*sgn(b*x + a) + 112*a*b^5*d^2*x^3*e^4*sgn(b*x + a) + 56
*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 16*a*b^5*d^4*x*e^2*sgn(b*x + a) + 2*a*b^5*d^5*e*sgn(b*x + a) + 210*a^2*b^4*x
^4*e^6*sgn(b*x + a) + 168*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 84*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 24*a^2*b^4*d^
3*x*e^3*sgn(b*x + a) + 3*a^2*b^4*d^4*e^2*sgn(b*x + a) + 224*a^3*b^3*x^3*e^6*sgn(b*x + a) + 112*a^3*b^3*d*x^2*e
^5*sgn(b*x + a) + 32*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 4*a^3*b^3*d^3*e^3*sgn(b*x + a) + 140*a^4*b^2*x^2*e^6*sgn
(b*x + a) + 40*a^4*b^2*d*x*e^5*sgn(b*x + a) + 5*a^4*b^2*d^2*e^4*sgn(b*x + a) + 48*a^5*b*x*e^6*sgn(b*x + a) + 6
*a^5*b*d*e^5*sgn(b*x + a) + 7*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^8