3.22.25 \(\int \frac {(a+b x) \sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) [2125]

Optimal. Leaf size=41 \[ \frac {2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 32} \begin {gather*} \frac {2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)*(d + e*x)^(3/2))/(3*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 784

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

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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.78 \begin {gather*} \frac {2 (a+b x) (d+e x)^{3/2}}{3 e \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 27, normalized size = 0.66

method	result	size
gosper	\(\frac {2 \left (b x +a \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e \sqrt {\left (b x +a \right )^{2}}}\)	\(27\)
default	\(\frac {2 \left (b x +a \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e \sqrt {\left (b x +a \right )^{2}}}\)	\(27\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{3 \left (b x +a \right ) e}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 12, normalized size = 0.29 \begin {gather*} \frac {2}{3} \, {\left (x e + d\right )}^{\frac {3}{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x*e + d)^(3/2)*e^(-1)

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Fricas [A]
time = 2.67, size = 12, normalized size = 0.29 \begin {gather*} \frac {2}{3} \, {\left (x e + d\right )}^{\frac {3}{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x*e + d)^(3/2)*e^(-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.56, size = 49, normalized size = 1.20 \begin {gather*} \frac {2}{3} \, {\left (3 \, \sqrt {x e + d} d \mathrm {sgn}\left (b x + a\right ) + {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*sqrt(x*e + d)*d*sgn(b*x + a) + ((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*sgn(b*x + a))*e^(-1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,\sqrt {d+e\,x}}{\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)^(1/2))/((a + b*x)^2)^(1/2),x)

[Out]

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

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