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

Optimal. Leaf size=39 \[ \frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)^3}{\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)^3}{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)^3 \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} \frac {(a+b x) (d+e x)^4}{4 e \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 31, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, e^{3} x^{4} + d e^{2} x^{3} + \frac {3}{2} \, d^{2} e x^{2} + d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*e^3*x^4 + d*e^2*x^3 + 3/2*d^2*e*x^2 + d^3*x

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giac [A]  time = 0.17, size = 18, normalized size = 0.46 \begin {gather*} \frac {1}{4} \, {\left (x e + d\right )}^{4} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/4*(x*e + d)^4*e^(-1)*sgn(b*x + a)

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maple [A]  time = 0.05, size = 47, normalized size = 1.21 \begin {gather*} \frac {\left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (b x +a \right ) x}{4 \sqrt {\left (b x +a \right )^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^3/((b*x+a)^2)^(1/2),x)

[Out]

1/4*x*(e^3*x^3+4*d*e^2*x^2+6*d^2*e*x+4*d^3)*(b*x+a)/((b*x+a)^2)^(1/2)

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maxima [B]  time = 0.53, size = 432, normalized size = 11.08 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{3}}{4 \, b} + \frac {13 \, a^{2} e^{3} x^{2}}{12 \, b^{2}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{3} x^{2}}{12 \, b^{2}} - \frac {13 \, a^{3} e^{3} x}{6 \, b^{3}} + \frac {a d^{3} \log \left (x + \frac {a}{b}\right )}{b} + \frac {a^{4} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{3}}{6 \, b^{4}} - \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a x^{2}}{6 \, b^{2}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} x^{2}}{2 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} + \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{3 \, b^{3}} - \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a x}{b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {2 \, {\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^3*x^3/b + 13/12*a^2*e^3*x^2/b^2 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*e^3
*x^2/b^2 - 13/6*a^3*e^3*x/b^3 + a*d^3*log(x + a/b)/b + a^4*e^3*log(x + a/b)/b^4 + 7/6*sqrt(b^2*x^2 + 2*a*b*x +
a^2)*a^3*e^3/b^4 - 5/6*(3*b*d*e^2 + a*e^3)*a*x^2/b^2 + 3/2*(b*d^2*e + a*d*e^2)*x^2/b + 1/3*(3*b*d*e^2 + a*e^3
)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 + 5/3*(3*b*d*e^2 + a*e^3)*a^2*x/b^3 - 3*(b*d^2*e + a*d*e^2)*a*x/b^2 -
(3*b*d*e^2 + a*e^3)*a^3*log(x + a/b)/b^4 + 3*(b*d^2*e + a*d*e^2)*a^2*log(x + a/b)/b^3 - (b*d^3 + 3*a*d^2*e)*a*
log(x + a/b)/b^2 - 2/3*(3*b*d*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^4 + (b*d^3 + 3*a*d^2*e)*sqrt(b^
2*x^2 + 2*a*b*x + a^2)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(d + e*x)^3)/((a + b*x)^2)^(1/2),x)

[Out]

int(((a + b*x)*(d + e*x)^3)/((a + b*x)^2)^(1/2), x)

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sympy [A]  time = 0.11, size = 32, normalized size = 0.82 \begin {gather*} d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**3/((b*x+a)**2)**(1/2),x)

[Out]

d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4

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