3.2025 \(\int \frac{(a+b x) (d+e x)^4}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{4 e^2 x (a+b x) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^4}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e (a+b x) (d+e x)^3}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (d+e x)^2 (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e (a+b x) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.106959, antiderivative size = 210, 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 = {768, 646, 43} \[ \frac{4 e^2 x (a+b x) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^4}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e (a+b x) (d+e x)^3}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 e (a+b x) (d+e x)^2 (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e (a+b x) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

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

Mathematica [A]  time = 0.0910808, size = 170, normalized size = 0.81 \[ \frac{6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )+3 a^3 b e^3 (4 d+3 e x)-3 a^4 e^4-2 a b^3 e \left (-9 d^2 e x-6 d^3+9 d e^2 x^2+e^3 x^3\right )-12 e (a+b x) (a e-b d)^3 \log (a+b x)+b^4 \left (18 d^2 e^2 x^2-3 d^4+6 d e^3 x^3+e^4 x^4\right )}{3 b^5 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(-3*a^4*e^4 + 3*a^3*b*e^3*(4*d + 3*e*x) + 6*a^2*b^2*e^2*(-3*d^2 - 4*d*e*x + e^2*x^2) - 2*a*b^3*e*(-6*d^3 - 9*d
^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(-(b*d) + a*e)^
3*(a + b*x)*Log[a + b*x])/(3*b^5*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.013, size = 321, normalized size = 1.5 \begin{align*} -{\frac{ \left ( -{x}^{4}{b}^{4}{e}^{4}+2\,{x}^{3}a{b}^{3}{e}^{4}-6\,{x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-36\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}+36\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}-12\,\ln \left ( bx+a \right ) x{b}^{4}{d}^{3}e-6\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+18\,{x}^{2}a{b}^{3}d{e}^{3}-18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-36\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+36\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-12\,\ln \left ( bx+a \right ) a{b}^{3}{d}^{3}e-9\,x{a}^{3}b{e}^{4}+24\,x{a}^{2}{b}^{2}d{e}^{3}-18\,xa{b}^{3}{d}^{2}{e}^{2}+3\,{a}^{4}{e}^{4}-12\,d{e}^{3}{a}^{3}b+18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-12\,a{b}^{3}{d}^{3}e+3\,{b}^{4}{d}^{4} \right ) \left ( bx+a \right ) ^{2}}{3\,{b}^{5}} \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)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

-1/3*(-x^4*b^4*e^4+2*x^3*a*b^3*e^4-6*x^3*b^4*d*e^3+12*ln(b*x+a)*x*a^3*b*e^4-36*ln(b*x+a)*x*a^2*b^2*d*e^3+36*ln
(b*x+a)*x*a*b^3*d^2*e^2-12*ln(b*x+a)*x*b^4*d^3*e-6*x^2*a^2*b^2*e^4+18*x^2*a*b^3*d*e^3-18*x^2*b^4*d^2*e^2+12*ln
(b*x+a)*a^4*e^4-36*ln(b*x+a)*a^3*b*d*e^3+36*ln(b*x+a)*a^2*b^2*d^2*e^2-12*ln(b*x+a)*a*b^3*d^3*e-9*x*a^3*b*e^4+2
4*x*a^2*b^2*d*e^3-18*x*a*b^3*d^2*e^2+3*a^4*e^4-12*d*e^3*a^3*b+18*a^2*b^2*d^2*e^2-12*a*b^3*d^3*e+3*b^4*d^4)*(b*
x+a)^2/b^5/((b*x+a)^2)^(3/2)

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Maxima [B]  time = 1.00897, size = 1220, normalized size = 5.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/3*e^4*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b) - 7/6*a*e^4*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 9/2*a^2*e^
4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) - 10*a^3*e^4*log(x + a/b)/((b^2)^(3/2)*b^2) - 15*a^5*e^4/((b^2)^(7/2
)*(x + a/b)^2) - 20*a^4*e^4*x/((b^2)^(5/2)*b*(x + a/b)^2) + 9*a^4*e^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5) - 1/
2*a*d^4/((b^2)^(3/2)*(x + a/b)^2) - 9/2*a^5*e^4/((b^2)^(3/2)*b^4*(x + a/b)^2) + 1/2*(4*b*d*e^3 + a*e^4)*x^3/(s
qrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 5/2*(4*b*d*e^3 + a*e^4)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) + 2*(3*b
*d^2*e^2 + 2*a*d*e^3)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 2*(2*b*d^3*e + 3*a*d^2*e^2)*log(x + a/b)/(b^2)
^(3/2) + 6*(4*b*d*e^3 + a*e^4)*a^2*log(x + a/b)/((b^2)^(3/2)*b^2) - 6*(3*b*d^2*e^2 + 2*a*d*e^3)*a*log(x + a/b)
/((b^2)^(3/2)*b) + 9*(4*b*d*e^3 + a*e^4)*a^4/((b^2)^(7/2)*(x + a/b)^2) - 9*(3*b*d^2*e^2 + 2*a*d*e^3)*a^3*b/((b
^2)^(7/2)*(x + a/b)^2) + 3*(2*b*d^3*e + 3*a*d^2*e^2)*a^2*b^2/((b^2)^(7/2)*(x + a/b)^2) - 12*(3*b*d^2*e^2 + 2*a
*d*e^3)*a^2*x/((b^2)^(5/2)*(x + a/b)^2) + 12*(4*b*d*e^3 + a*e^4)*a^3*x/((b^2)^(5/2)*b*(x + a/b)^2) + 4*(2*b*d^
3*e + 3*a*d^2*e^2)*a*b*x/((b^2)^(5/2)*(x + a/b)^2) - 5*(4*b*d*e^3 + a*e^4)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*
b^5) + 4*(3*b*d^2*e^2 + 2*a*d*e^3)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - (b*d^4 + 4*a*d^3*e)/(sqrt(b^2*x^2
+ 2*a*b*x + a^2)*b^2) + 5/2*(4*b*d*e^3 + a*e^4)*a^4/((b^2)^(3/2)*b^4*(x + a/b)^2) - 2*(3*b*d^2*e^2 + 2*a*d*e^
3)*a^3/((b^2)^(3/2)*b^3*(x + a/b)^2) + 1/2*(b*d^4 + 4*a*d^3*e)*a/((b^2)^(3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 1.49884, size = 540, normalized size = 2.57 \begin{align*} \frac{b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \,{\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**4/((a + b*x)**2)**(3/2), x)

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Giac [A]  time = 1.23596, size = 320, normalized size = 1.52 \begin{align*} \frac{1}{3} \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}{\left (x{\left (\frac{x e^{4}}{b^{3}} + \frac{2 \,{\left (3 \, b^{13} d e^{3} - 2 \, a b^{12} e^{4}\right )}}{b^{16}}\right )} + \frac{18 \, b^{13} d^{2} e^{2} - 30 \, a b^{12} d e^{3} + 13 \, a^{2} b^{11} e^{4}}{b^{16}}\right )} - \frac{4 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | -3 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{2} a b - a^{3} b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{3}{\left | b \right |} - 3 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )} a^{2}{\left | b \right |} \right |}\right )}{3 \, b^{4}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*(x*(x*e^4/b^3 + 2*(3*b^13*d*e^3 - 2*a*b^12*e^4)/b^16) + (18*b^13*d^2*e^2 - 3
0*a*b^12*d*e^3 + 13*a^2*b^11*e^4)/b^16) - 4/3*(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*log(abs(
-3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2))^2*a*b - a^3*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2))^3*abs
(b) - 3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2))*a^2*abs(b)))/(b^4*abs(b))