3.6.70 \(\int \frac {(d+e x)^3}{(a+c x^2)^{3/2}} \, dx\) [570]

Optimal. Leaf size=106 \[ -\frac {(a e-c d x) (d+e x)^2}{a c \sqrt {a+c x^2}}-\frac {e \left (2 \left (c d^2-a e^2\right )+c d e x\right ) \sqrt {a+c x^2}}{a c^2}+\frac {3 d e^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {753, 794, 223, 212} \begin {gather*} \frac {3 d e^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}-\frac {e \sqrt {a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}-\frac {(d+e x)^2 (a e-c d x)}{a c \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + c*x^2)^(3/2),x]

[Out]

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

Rule 212

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

Rule 223

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 753

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.40, size = 90, normalized size = 0.85 \begin {gather*} \frac {2 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )}{a c^2 \sqrt {a+c x^2}}-\frac {3 d e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.44, size = 115, normalized size = 1.08

method	result	size
risch	\(\frac {e^{3} \sqrt {c \,x^{2}+a}}{c^{2}}-\frac {3 d \,e^{2} x}{c \sqrt {c \,x^{2}+a}}+\frac {3 d \,e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}+\frac {a \,e^{3}}{c^{2} \sqrt {c \,x^{2}+a}}-\frac {3 d^{2} e}{c \sqrt {c \,x^{2}+a}}+\frac {d^{3} x}{a \sqrt {c \,x^{2}+a}}\)	\(114\)
default	\(e^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )+3 d \,e^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )-\frac {3 d^{2} e}{c \sqrt {c \,x^{2}+a}}+\frac {d^{3} x}{a \sqrt {c \,x^{2}+a}}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 107, normalized size = 1.01 \begin {gather*} \frac {d^{3} x}{\sqrt {c x^{2} + a} a} + \frac {x^{2} e^{3}}{\sqrt {c x^{2} + a} c} - \frac {3 \, d x e^{2}}{\sqrt {c x^{2} + a} c} + \frac {3 \, d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{c^{\frac {3}{2}}} - \frac {3 \, d^{2} e}{\sqrt {c x^{2} + a} c} + \frac {2 \, a e^{3}}{\sqrt {c x^{2} + a} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 7.75, size = 232, normalized size = 2.19 \begin {gather*} \left [\frac {3 \, {\left (a c d x^{2} + a^{2} d\right )} \sqrt {c} e^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (c^{2} d^{3} x - 3 \, a c d x e^{2} - 3 \, a c d^{2} e + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac {3 \, {\left (a c d x^{2} + a^{2} d\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) e^{2} - {\left (c^{2} d^{3} x - 3 \, a c d x e^{2} - 3 \, a c d^{2} e + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{a c^{3} x^{2} + a^{2} c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.63, size = 100, normalized size = 0.94 \begin {gather*} -\frac {3 \, d e^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {3}{2}}} + \frac {x {\left (\frac {x e^{3}}{c} + \frac {c^{3} d^{3} - 3 \, a c^{2} d e^{2}}{a c^{3}}\right )} - \frac {3 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}}{a c^{3}}}{\sqrt {c x^{2} + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(a + c*x^2)^(3/2),x)

[Out]

int((d + e*x)^3/(a + c*x^2)^(3/2), x)

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