3.6.79 \(\int \frac {(d+e x)^3}{(a+c x^2)^{5/2}} \, dx\) [579]

Optimal. Leaf size=79 \[ -\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {737, 651} \begin {gather*} -\frac {2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(a + c*x^2)^(5/2),x]

[Out]

-1/3*((a*e - c*d*x)*(d + e*x)^2)/(a*c*(a + c*x^2)^(3/2)) - (2*(c*d^2 + a*e^2)*(a*e - c*d*x))/(3*a^2*c^2*Sqrt[a
+ c*x^2])

Rule 651

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 737

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[(2*p + 3)*((c*d^2 + a*e^2)/(2*a*c*(p + 1))), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\left (2 \left (c d^2+a e^2\right )\right ) \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 78, normalized size = 0.99 \begin {gather*} \frac {-2 a^3 e^3+2 c^3 d^3 x^3-3 a^2 c e \left (d^2+e^2 x^2\right )+3 a c^2 d x \left (d^2+e^2 x^2\right )}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/(a + c*x^2)^(5/2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(71)=142\).
time = 0.42, size = 151, normalized size = 1.91

method	result	size
gosper	\(-\frac {-3 a \,c^{2} d \,e^{2} x^{3}-2 c^{3} d^{3} x^{3}+3 e^{3} x^{2} a^{2} c -3 d^{3} x a \,c^{2}+2 e^{3} a^{3}+3 d^{2} e \,a^{2} c}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2}}\)	\(83\)
trager	\(-\frac {-3 a \,c^{2} d \,e^{2} x^{3}-2 c^{3} d^{3} x^{3}+3 e^{3} x^{2} a^{2} c -3 d^{3} x a \,c^{2}+2 e^{3} a^{3}+3 d^{2} e \,a^{2} c}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2}}\)	\(83\)
default	\(e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+3 d \,e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )-\frac {d^{2} e}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+d^{3} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 130, normalized size = 1.65 \begin {gather*} \frac {2 \, d^{3} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{3} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {x^{2} e^{3}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {d x e^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {d x e^{2}}{\sqrt {c x^{2} + a} a c} - \frac {d^{2} e}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {2 \, a e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.34, size = 106, normalized size = 1.34 \begin {gather*} \frac {{\left (2 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d x^{3} e^{2} + 3 \, a c^{2} d^{3} x - 3 \, a^{2} c d^{2} e - {\left (3 \, a^{2} c x^{2} + 2 \, a^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} 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)^(5/2),x, algorithm="fricas")

[Out]

1/3*(2*c^3*d^3*x^3 + 3*a*c^2*d*x^3*e^2 + 3*a*c^2*d^3*x - 3*a^2*c*d^2*e - (3*a^2*c*x^2 + 2*a^3)*e^3)*sqrt(c*x^2
+ a)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*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 {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**3/(a + c*x**2)**(5/2), x)

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Giac [A]
time = 3.82, size = 88, normalized size = 1.11 \begin {gather*} \frac {{\left (\frac {3 \, d^{3}}{a} - x {\left (\frac {3 \, e^{3}}{c} - \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )}\right )} x - \frac {3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.51, size = 82, normalized size = 1.04 \begin {gather*} -\frac {2\,a^3\,e^3+3\,a^2\,c\,d^2\,e+3\,a^2\,c\,e^3\,x^2-3\,a\,c^2\,d^3\,x-3\,a\,c^2\,d\,e^2\,x^3-2\,c^3\,d^3\,x^3}{3\,a^2\,c^2\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3/(a + c*x^2)^(5/2),x)

[Out]

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

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