3.1734 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \]

[Out]

-((b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) + ((b*d - a*e)^2*(4*b
*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7) - (b*(b*d - a*e)*(2*b*B*d
- A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)*(d + e*x)^6) + (b^2*(4*b*B*d - A*b*e - 3*a*B
*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^5) - (b^3*B*Sqrt[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 = 0.181199, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{5 e^5 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (a+b x) (d+e x)^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x) (d+e x)^7}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4} \]

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)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)*(d + e*x)^8) + ((b*d - a*e)^2*(4*b
*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^7) - (b*(b*d - a*e)*(2*b*B*d
- A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)*(d + e*x)^6) + (b^2*(4*b*B*d - A*b*e - 3*a*B
*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^5) - (b^3*B*Sqrt[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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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{\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^9} \, 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 (-B d+A e)}{e^4 (d+e x)^9}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^8}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^7}+\frac{b^5 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^6}+\frac{b^6 B}{e^4 (d+e x)^5}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^3 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x) (d+e x)^8}+\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}-\frac{b (b d-a e) (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^6}+\frac{b^2 (4 b B d-A b e-3 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}-\frac{b^3 B \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 = 0.11564, size = 229, normalized size = 0.77 \[ -\frac{\sqrt{(a+b x)^2} \left (5 a^2 b e^2 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+5 a^3 e^3 (7 A e+B (d+8 e x))+a b^2 e \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+b^3 \left (A e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+B \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )\right )}{280 e^5 (a+b x) (d+e x)^8} \]

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]*(5*a^3*e^3*(7*A*e + B*(d + 8*e*x)) + 5*a^2*b*e^2*(3*A*e*(d + 8*e*x) + B*(d^2 + 8*d*e*x + 2
8*e^2*x^2)) + a*b^2*e*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3))
+ b^3*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(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 = 0.01, size = 315, normalized size = 1.1 \begin{align*} -{\frac{70\,B{x}^{4}{b}^{3}{e}^{4}+56\,A{x}^{3}{b}^{3}{e}^{4}+168\,B{x}^{3}a{b}^{2}{e}^{4}+56\,B{x}^{3}{b}^{3}d{e}^{3}+140\,A{x}^{2}a{b}^{2}{e}^{4}+28\,A{x}^{2}{b}^{3}d{e}^{3}+140\,B{x}^{2}{a}^{2}b{e}^{4}+84\,B{x}^{2}a{b}^{2}d{e}^{3}+28\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+120\,Ax{a}^{2}b{e}^{4}+40\,Axa{b}^{2}d{e}^{3}+8\,Ax{b}^{3}{d}^{2}{e}^{2}+40\,Bx{a}^{3}{e}^{4}+40\,Bx{a}^{2}bd{e}^{3}+24\,Bxa{b}^{2}{d}^{2}{e}^{2}+8\,Bx{b}^{3}{d}^{3}e+35\,A{a}^{3}{e}^{4}+15\,Ad{e}^{3}{a}^{2}b+5\,Aa{b}^{2}{d}^{2}{e}^{2}+A{b}^{3}{d}^{3}e+5\,Bd{e}^{3}{a}^{3}+5\,B{a}^{2}b{d}^{2}{e}^{2}+3\,Ba{b}^{2}{d}^{3}e+B{b}^{3}{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*b^3*e^4*x^4+56*A*b^3*e^4*x^3+168*B*a*b^2*e^4*x^3+56*B*b^3*d*e^3*x^3+140*A*a*b^2*e^4*x^2+28*A*
b^3*d*e^3*x^2+140*B*a^2*b*e^4*x^2+84*B*a*b^2*d*e^3*x^2+28*B*b^3*d^2*e^2*x^2+120*A*a^2*b*e^4*x+40*A*a*b^2*d*e^3
*x+8*A*b^3*d^2*e^2*x+40*B*a^3*e^4*x+40*B*a^2*b*d*e^3*x+24*B*a*b^2*d^2*e^2*x+8*B*b^3*d^3*e*x+35*A*a^3*e^4+15*A*
a^2*b*d*e^3+5*A*a*b^2*d^2*e^2+A*b^3*d^3*e+5*B*a^3*d*e^3+5*B*a^2*b*d^2*e^2+3*B*a*b^2*d^3*e+B*b^3*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 = 1.65188, size = 701, normalized size = 2.35 \begin{align*} -\frac{70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \,{\left (B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \,{\left (B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \,{\left (B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} 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*b^3*e^4*x^4 + B*b^3*d^4 + 35*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e + 5*(B*a^2*b + A*a*b^2)*d^2*e^
2 + 5*(B*a^3 + 3*A*a^2*b)*d*e^3 + 56*(B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 28*(B*b^3*d^2*e^2 + (3*B*a*
b^2 + A*b^3)*d*e^3 + 5*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 8*(B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 5*(B*a^2*
b + A*a*b^2)*d*e^3 + 5*(B*a^3 + 3*A*a^2*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 = 1.14428, size = 574, normalized size = 1.93 \begin{align*} -\frac{{\left (70 \, B b^{3} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, B b^{3} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, B b^{3} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, B b^{3} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) + 168 \, B a b^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, A b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, B a b^{2} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, A b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 24 \, B a b^{2} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, A b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) + A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 140 \, B a^{2} b x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 140 \, A a b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 40 \, B a^{2} b d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 40 \, A a b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 40 \, B a^{3} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 120 \, A a^{2} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 5 \, B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 35 \, A a^{3} 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*b^3*x^4*e^4*sgn(b*x + a) + 56*B*b^3*d*x^3*e^3*sgn(b*x + a) + 28*B*b^3*d^2*x^2*e^2*sgn(b*x + a) +
8*B*b^3*d^3*x*e*sgn(b*x + a) + B*b^3*d^4*sgn(b*x + a) + 168*B*a*b^2*x^3*e^4*sgn(b*x + a) + 56*A*b^3*x^3*e^4*sg
n(b*x + a) + 84*B*a*b^2*d*x^2*e^3*sgn(b*x + a) + 28*A*b^3*d*x^2*e^3*sgn(b*x + a) + 24*B*a*b^2*d^2*x*e^2*sgn(b*
x + a) + 8*A*b^3*d^2*x*e^2*sgn(b*x + a) + 3*B*a*b^2*d^3*e*sgn(b*x + a) + A*b^3*d^3*e*sgn(b*x + a) + 140*B*a^2*
b*x^2*e^4*sgn(b*x + a) + 140*A*a*b^2*x^2*e^4*sgn(b*x + a) + 40*B*a^2*b*d*x*e^3*sgn(b*x + a) + 40*A*a*b^2*d*x*e
^3*sgn(b*x + a) + 5*B*a^2*b*d^2*e^2*sgn(b*x + a) + 5*A*a*b^2*d^2*e^2*sgn(b*x + a) + 40*B*a^3*x*e^4*sgn(b*x + a
) + 120*A*a^2*b*x*e^4*sgn(b*x + a) + 5*B*a^3*d*e^3*sgn(b*x + a) + 15*A*a^2*b*d*e^3*sgn(b*x + a) + 35*A*a^3*e^4
*sgn(b*x + a))*e^(-5)/(x*e + d)^8