3.5.48 \(\int (d+e x) \sqrt {a+c x^2} \, dx\)

Optimal. Leaf size=67 \[ \frac {1}{2} d x \sqrt {a+c x^2}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c} \]

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Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \begin {gather*} \frac {1}{2} d x \sqrt {a+c x^2}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*Sqrt[a + c*x^2],x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (d+e x) \sqrt {a+c x^2} \, dx &=\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+d \int \sqrt {a+c x^2} \, dx\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {1}{2} (a d) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {1}{2} (a d) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {1}{2} d x \sqrt {a+c x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 c}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^2} (2 a e+c x (3 d+2 e x))+3 a \sqrt {c} d \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(2*a*e + c*x*(3*d + 2*e*x)) + 3*a*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c)

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IntegrateAlgebraic [A]  time = 0.25, size = 68, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+c x^2} \left (2 a e+3 c d x+2 c e x^2\right )}{6 c}-\frac {a d \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)*Sqrt[a + c*x^2],x]

[Out]

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

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fricas [A]  time = 0.43, size = 128, normalized size = 1.91 \begin {gather*} \left [\frac {3 \, a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt {c x^{2} + a}}{12 \, c}, -\frac {3 \, a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c e x^{2} + 3 \, c d x + 2 \, a e\right )} \sqrt {c x^{2} + a}}{6 \, c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*a*sqrt(c)*d*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(2*c*e*x^2 + 3*c*d*x + 2*a*e)*sqrt(c*
x^2 + a))/c, -1/6*(3*a*sqrt(-c)*d*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (2*c*e*x^2 + 3*c*d*x + 2*a*e)*sqrt(c*x^
2 + a))/c]

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giac [A]  time = 0.19, size = 57, normalized size = 0.85 \begin {gather*} -\frac {a d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, \sqrt {c}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, x e + 3 \, d\right )} x + \frac {2 \, a e}{c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*d*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/6*sqrt(c*x^2 + a)*((2*x*e + 3*d)*x + 2*a*e/c)

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maple [A]  time = 0.04, size = 53, normalized size = 0.79 \begin {gather*} \frac {a d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}+\frac {\sqrt {c \,x^{2}+a}\, d x}{2}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e}{3 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 45, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + a} d x + \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2 + a)*d*x + 1/2*a*d*arcsinh(c*x/sqrt(a*c))/sqrt(c) + 1/3*(c*x^2 + a)^(3/2)*e/c

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mupad [B]  time = 0.53, size = 52, normalized size = 0.78 \begin {gather*} \frac {e\,{\left (c\,x^2+a\right )}^{3/2}}{3\,c}+\frac {d\,x\,\sqrt {c\,x^2+a}}{2}+\frac {a\,d\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{2\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(1/2)*(d + e*x),x)

[Out]

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

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sympy [A]  time = 3.34, size = 70, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a} d x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 \sqrt {c}} + e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*d*x*sqrt(1 + c*x**2/a)/2 + a*d*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + e*Piecewise((sqrt(a)*x**2/2, Eq(
c, 0)), ((a + c*x**2)**(3/2)/(3*c), True))

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