3.1978 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=238 \[ -\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (3 b d-4 a e)}{e^4 (a+b x)}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5 (a+b x) (d+e x)^2}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^5 (a+b x)}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.149063, antiderivative size = 238, 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} \[ -\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (3 b d-4 a e)}{e^4 (a+b x)}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5 (a+b x) (d+e x)^2}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^5 (a+b x)}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.129222, size = 185, normalized size = 0.78 \[ \frac{\sqrt{(a+b x)^2} \left (6 a^2 b^2 d e^2 (3 d+4 e x)-4 a^3 b e^3 (d+2 e x)-a^4 e^4+4 a b^3 e \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+12 b^2 (d+e x)^2 (b d-a e)^2 \log (d+e x)+b^4 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )\right )}{2 e^5 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(-(a^4*e^4) - 4*a^3*b*e^3*(d + 2*e*x) + 6*a^2*b^2*d*e^2*(3*d + 4*e*x) + 4*a*b^3*e*(-5*d^3 -
4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + b^4*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 12*
b^2*(b*d - a*e)^2*(d + e*x)^2*Log[d + e*x]))/(2*e^5*(a + b*x)*(d + e*x)^2)

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Maple [A]  time = 0.015, size = 350, normalized size = 1.5 \begin{align*}{\frac{{x}^{4}{b}^{4}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+8\,{x}^{3}a{b}^{3}{e}^{4}-4\,{x}^{3}{b}^{4}d{e}^{3}+24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-48\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+24\,\ln \left ( ex+d \right ) x{b}^{4}{d}^{3}e+16\,{x}^{2}a{b}^{3}d{e}^{3}-11\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-24\,\ln \left ( ex+d \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}-8\,x{a}^{3}b{e}^{4}+24\,x{a}^{2}{b}^{2}d{e}^{3}-16\,xa{b}^{3}{d}^{2}{e}^{2}+2\,x{b}^{4}{d}^{3}e-{a}^{4}{e}^{4}-4\,d{e}^{3}{a}^{3}b+18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-20\,a{b}^{3}{d}^{3}e+7\,{b}^{4}{d}^{4}}{2\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{2}} \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)^3,x)

[Out]

1/2*((b*x+a)^2)^(3/2)*(x^4*b^4*e^4+12*ln(e*x+d)*x^2*a^2*b^2*e^4-24*ln(e*x+d)*x^2*a*b^3*d*e^3+12*ln(e*x+d)*x^2*
b^4*d^2*e^2+8*x^3*a*b^3*e^4-4*x^3*b^4*d*e^3+24*ln(e*x+d)*x*a^2*b^2*d*e^3-48*ln(e*x+d)*x*a*b^3*d^2*e^2+24*ln(e*
x+d)*x*b^4*d^3*e+16*x^2*a*b^3*d*e^3-11*x^2*b^4*d^2*e^2+12*ln(e*x+d)*a^2*b^2*d^2*e^2-24*ln(e*x+d)*a*b^3*d^3*e+1
2*ln(e*x+d)*b^4*d^4-8*x*a^3*b*e^4+24*x*a^2*b^2*d*e^3-16*x*a*b^3*d^2*e^2+2*x*b^4*d^3*e-a^4*e^4-4*d*e^3*a^3*b+18
*a^2*b^2*d^2*e^2-20*a*b^3*d^3*e+7*b^4*d^4)/(b*x+a)^3/e^5/(e*x+d)^2

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51229, size = 586, normalized size = 2.46 \begin{align*} \frac{b^{4} e^{4} x^{4} + 7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} - 4 \,{\left (b^{4} d e^{3} - 2 \, a b^{3} e^{4}\right )} x^{3} -{\left (11 \, b^{4} d^{2} e^{2} - 16 \, a b^{3} d e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 12 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 2 \, a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/2*(b^4*e^4*x^4 + 7*b^4*d^4 - 20*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 - a^4*e^4 - 4*(b^4*d*e^3 -
2*a*b^3*e^4)*x^3 - (11*b^4*d^2*e^2 - 16*a*b^3*d*e^3)*x^2 + 2*(b^4*d^3*e - 8*a*b^3*d^2*e^2 + 12*a^2*b^2*d*e^3 -
4*a^3*b*e^4)*x + 12*(b^4*d^4 - 2*a*b^3*d^3*e + a^2*b^2*d^2*e^2 + (b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*
x^2 + 2*(b^4*d^3*e - 2*a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(e*x + d))/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{3}}\, dx \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)**3,x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(3/2)/(d + e*x)**3, x)

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Giac [A]  time = 1.14907, size = 358, normalized size = 1.5 \begin{align*} 6 \,{\left (b^{4} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{3} d e \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{2} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{4} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 6 \, b^{4} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, a b^{3} x e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} + \frac{{\left (7 \, b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) - 20 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) - a^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 8 \,{\left (b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) - a^{3} b e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

6*(b^4*d^2*sgn(b*x + a) - 2*a*b^3*d*e*sgn(b*x + a) + a^2*b^2*e^2*sgn(b*x + a))*e^(-5)*log(abs(x*e + d)) + 1/2*
(b^4*x^2*e^3*sgn(b*x + a) - 6*b^4*d*x*e^2*sgn(b*x + a) + 8*a*b^3*x*e^3*sgn(b*x + a))*e^(-6) + 1/2*(7*b^4*d^4*s
gn(b*x + a) - 20*a*b^3*d^3*e*sgn(b*x + a) + 18*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) - a^4
*e^4*sgn(b*x + a) + 8*(b^4*d^3*e*sgn(b*x + a) - 3*a*b^3*d^2*e^2*sgn(b*x + a) + 3*a^2*b^2*d*e^3*sgn(b*x + a) -
a^3*b*e^4*sgn(b*x + a))*x)*e^(-5)/(x*e + d)^2