3.2015 \(\int \frac{(a+b x) (d+e x)^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)
Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
((a + b*x)*(d + e*x)^5)/(5*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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Rubi [A] time = 0.0289572, antiderivative size = 39, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used =
{770, 21, 32} \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)*(d + e*x)^4)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
((a + b*x)*(d + e*x)^5)/(5*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{(a+b x) (d+e x)^4}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int (d+e x)^4 \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(a+b x) (d+e x)^5}{5 e \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0203262, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)*(d + e*x)^4)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
((a + b*x)*(d + e*x)^5)/(5*e*Sqrt[(a + b*x)^2])
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Maple [B] time = 0.003, size = 58, normalized size = 1.5 \begin{align*}{\frac{x \left ({e}^{4}{x}^{4}+5\,{e}^{3}{x}^{3}d+10\,{e}^{2}{x}^{2}{d}^{2}+10\,e{d}^{3}x+5\,{d}^{4} \right ) \left ( bx+a \right ) }{5}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^4/((b*x+a)^2)^(1/2),x)
[Out]
1/5*x*(e^4*x^4+5*d*e^3*x^3+10*d^2*e^2*x^2+10*d^3*e*x+5*d^4)*(b*x+a)/((b*x+a)^2)^(1/2)
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Maxima [B] time = 1.02907, size = 1129, normalized size = 28.95 \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*x+a)^2)^(1/2),x, algorithm="maxima")
[Out]
1/5*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^4*x^4/b - 77/30*a^5*e^4*log(x + a/b)/(b^2)^(5/2) + 77/30*a^4*e^4*x/((b^2)^
(3/2)*b) - 77/60*a^3*e^4*x^2/(sqrt(b^2)*b^2) - 9/20*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*e^4*x^3/b^2 + a*sqrt(b^(-2
))*d^4*log(x + a/b) + 47/30*a^5*sqrt(b^(-2))*e^4*log(x + a/b)/b^4 + 47/60*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*e^
4*x^2/b^3 - 47/30*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4*e^4/b^5 + 13/6*(4*b*d*e^3 + a*e^4)*a^4*log(x + a/b)/(b^2)^
(5/2) - 10/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a^3*b*log(x + a/b)/(b^2)^(5/2) + 2*(2*b*d^3*e + 3*a*d^2*e^2)*a^2*b^2*lo
g(x + a/b)/(b^2)^(5/2) + 10/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a^2*x/(b^2)^(3/2) - 13/6*(4*b*d*e^3 + a*e^4)*a^3*x/((b
^2)^(3/2)*b) - 2*(2*b*d^3*e + 3*a*d^2*e^2)*a*b*x/(b^2)^(3/2) + (2*b*d^3*e + 3*a*d^2*e^2)*x^2/sqrt(b^2) + 13/12
*(4*b*d*e^3 + a*e^4)*a^2*x^2/(sqrt(b^2)*b^2) - 5/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a*x^2/(sqrt(b^2)*b) + 1/4*(4*b*d*
e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^3/b^2 - 7/6*(4*b*d*e^3 + a*e^4)*a^4*sqrt(b^(-2))*log(x + a/b)/b^4
+ 4/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a^3*sqrt(b^(-2))*log(x + a/b)/b^3 - (b*d^4 + 4*a*d^3*e)*a*sqrt(b^(-2))*log(x
+ a/b)/b - 7/12*(4*b*d*e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x^2/b^3 + 2/3*(3*b*d^2*e^2 + 2*a*d*e^3)*sq
rt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 + 7/6*(4*b*d*e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3/b^5 - 4/3*(3*b
*d^2*e^2 + 2*a*d*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*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
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Fricas [A] time = 1.46195, size = 85, normalized size = 2.18 \begin{align*} \frac{1}{5} \, e^{4} x^{5} + d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{3} + 2 \, d^{3} e x^{2} + d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^4/((b*x+a)^2)^(1/2),x, algorithm="fricas")
[Out]
1/5*e^4*x^5 + d*e^3*x^4 + 2*d^2*e^2*x^3 + 2*d^3*e*x^2 + d^4*x
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Sympy [A] time = 0.116721, size = 42, normalized size = 1.08 \begin{align*} d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac{e^{4} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**4/((b*x+a)**2)**(1/2),x)
[Out]
d**4*x + 2*d**3*e*x**2 + 2*d**2*e**2*x**3 + d*e**3*x**4 + e**4*x**5/5
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Giac [A] time = 1.12199, size = 24, normalized size = 0.62 \begin{align*} \frac{1}{5} \,{\left (x e + d\right )}^{5} e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^4/((b*x+a)^2)^(1/2),x, algorithm="giac")
[Out]
1/5*(x*e + d)^5*e^(-1)*sgn(b*x + a)