3.1984 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=252 \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{8 e^5 (a+b x) (d+e x)^8} \]

[Out]

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

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Rubi [A]  time = 0.130678, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^6}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{8 e^5 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

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

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 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{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(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 )^3}{(d+e x)^9} \, dx}{b^2 \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)^4}{(d+e x)^9} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^9}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac{b^4}{e^4 (d+e x)^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\frac{4 b (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac{b^2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^6}+\frac{4 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0615593, size = 162, normalized size = 0.64 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+20 a^3 b e^3 (d+8 e x)+35 a^4 e^4+4 a b^3 e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )}{280 e^5 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a
*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(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)))/(280*e^5*(a + b*x)*(d + e*x)^8)

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Maple [A]  time = 0.007, size = 201, normalized size = 0.8 \begin{align*} -{\frac{70\,{x}^{4}{b}^{4}{e}^{4}+224\,{x}^{3}a{b}^{3}{e}^{4}+56\,{x}^{3}{b}^{4}d{e}^{3}+280\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}+28\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+160\,x{a}^{3}b{e}^{4}+80\,x{a}^{2}{b}^{2}d{e}^{3}+32\,xa{b}^{3}{d}^{2}{e}^{2}+8\,x{b}^{4}{d}^{3}e+35\,{a}^{4}{e}^{4}+20\,d{e}^{3}{a}^{3}b+10\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{280\,{e}^{5} \left ( ex+d \right ) ^{8} \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*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^9,x)

[Out]

-1/280/e^5*(70*b^4*e^4*x^4+224*a*b^3*e^4*x^3+56*b^4*d*e^3*x^3+280*a^2*b^2*e^4*x^2+112*a*b^3*d*e^3*x^2+28*b^4*d
^2*e^2*x^2+160*a^3*b*e^4*x+80*a^2*b^2*d*e^3*x+32*a*b^3*d^2*e^2*x+8*b^4*d^3*e*x+35*a^4*e^4+20*a^3*b*d*e^3+10*a^
2*b^2*d^2*e^2+4*a*b^3*d^3*e+b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^8/(b*x+a)^3

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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)^(3/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50192, size = 540, normalized size = 2.14 \begin{align*} -\frac{70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \,{\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\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)^(3/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*
d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*
e^2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^
4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

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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)**(3/2)/(e*x+d)**9,x)

[Out]

Timed out

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Giac [A]  time = 1.15452, size = 356, normalized size = 1.41 \begin{align*} -\frac{{\left (70 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 224 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 112 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 32 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 280 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 80 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 160 \, a^{3} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{280 \,{\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)^(3/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(70*b^4*x^4*e^4*sgn(b*x + a) + 56*b^4*d*x^3*e^3*sgn(b*x + a) + 28*b^4*d^2*x^2*e^2*sgn(b*x + a) + 8*b^4*
d^3*x*e*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 224*a*b^3*x^3*e^4*sgn(b*x + a) + 112*a*b^3*d*x^2*e^3*sgn(b*x + a
) + 32*a*b^3*d^2*x*e^2*sgn(b*x + a) + 4*a*b^3*d^3*e*sgn(b*x + a) + 280*a^2*b^2*x^2*e^4*sgn(b*x + a) + 80*a^2*b
^2*d*x*e^3*sgn(b*x + a) + 10*a^2*b^2*d^2*e^2*sgn(b*x + a) + 160*a^3*b*x*e^4*sgn(b*x + a) + 20*a^3*b*d*e^3*sgn(
b*x + a) + 35*a^4*e^4*sgn(b*x + a))*e^(-5)/(x*e + d)^8