3.8.22 \(\int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=78 \[ -\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]

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Rubi [A]  time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {657, 649} \begin {gather*} -\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2])/(3*c*e)

Rule 649

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

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx &=-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e}+\frac {1}{3} (4 d) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 47, normalized size = 0.60 \begin {gather*} -\frac {2 (d-e x) \sqrt {d+e x} (5 d+e x)}{3 e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d - e*x)*Sqrt[d + e*x]*(5*d + e*x))/(3*e*Sqrt[c*(d^2 - e^2*x^2)])

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IntegrateAlgebraic [A]  time = 0.12, size = 50, normalized size = 0.64 \begin {gather*} -\frac {2 (5 d+e x) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{3 c e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(5*d + e*x)*Sqrt[2*c*d*(d + e*x) - c*(d + e*x)^2])/(3*c*e*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 46, normalized size = 0.59 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + 5 \, d\right )} \sqrt {e x + d}}{3 \, {\left (c e^{2} x + c d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(-c*e^2*x^2 + c*d^2)*(e*x + 5*d)*sqrt(e*x + d)/(c*e^2*x + c*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(3/2)/sqrt(-c*e^2*x^2 + c*d^2), x)

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maple [A]  time = 0.05, size = 43, normalized size = 0.55 \begin {gather*} -\frac {2 \left (-e x +d \right ) \left (e x +5 d \right ) \sqrt {e x +d}}{3 \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-e*x+d)*(e*x+5*d)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1/2)

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maxima [A]  time = 1.50, size = 34, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left (e^{2} x^{2} + 4 \, d e x - 5 \, d^{2}\right )}}{3 \, \sqrt {-e x + d} \sqrt {c} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(e^2*x^2 + 4*d*e*x - 5*d^2)/(sqrt(-e*x + d)*sqrt(c)*e)

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mupad [B]  time = 0.57, size = 61, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {10\,d\,\sqrt {d+e\,x}}{3\,c\,e^2}+\frac {2\,x\,\sqrt {d+e\,x}}{3\,c\,e}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((10*d*(d + e*x)^(1/2))/(3*c*e^2) + (2*x*(d + e*x)^(1/2))/(3*c*e)))/(x + d/e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

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

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