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

Optimal. Leaf size=119 \[ -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(315*c*e*(d + e*x)^(5/2)) - (16*d*(c*d^2 - c*e^2*x^2)^(5/2))/(63*c*e*(d +
e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(9*c*e*Sqrt[d + e*x])

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 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}+\frac {1}{9} (8 d) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}+\frac {1}{63} \left (32 d^2\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 62, normalized size = 0.52 \begin {gather*} -\frac {2 c (d-e x)^2 \left (107 d^2+110 d e x+35 e^2 x^2\right ) \sqrt {c \left (d^2-e^2 x^2\right )}}{315 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(107*d^2 + 110*d*e*x + 35*e^2*x^2))/(315*e*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.47, size = 92, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt {2 c d (d+e x)-c (d+e x)^2} \left (128 c d^4+32 c d^3 (d+e x)+12 c d^2 (d+e x)^2-100 c d (d+e x)^3+35 c (d+e x)^4\right )}{315 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2*c*d*(d + e*x) - c*(d + e*x)^2]*(128*c*d^4 + 32*c*d^3*(d + e*x) + 12*c*d^2*(d + e*x)^2 - 100*c*d*(d
+ e*x)^3 + 35*c*(d + e*x)^4))/(315*e*Sqrt[d + e*x])

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fricas [A]  time = 0.40, size = 83, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (35 \, c e^{4} x^{4} + 40 \, c d e^{3} x^{3} - 78 \, c d^{2} e^{2} x^{2} - 104 \, c d^{3} e x + 107 \, c d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(35*c*e^4*x^4 + 40*c*d*e^3*x^3 - 78*c*d^2*e^2*x^2 - 104*c*d^3*e*x + 107*c*d^4)*sqrt(-c*e^2*x^2 + c*d^2)
*sqrt(e*x + d)/(e^2*x + d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 55, normalized size = 0.46 \begin {gather*} -\frac {2 \left (-e x +d \right ) \left (35 e^{2} x^{2}+110 d e x +107 d^{2}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(-e*x+d)*(35*e^2*x^2+110*d*e*x+107*d^2)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

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maxima [A]  time = 1.57, size = 82, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (35 \, c^{\frac {3}{2}} e^{4} x^{4} + 40 \, c^{\frac {3}{2}} d e^{3} x^{3} - 78 \, c^{\frac {3}{2}} d^{2} e^{2} x^{2} - 104 \, c^{\frac {3}{2}} d^{3} e x + 107 \, c^{\frac {3}{2}} d^{4}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(35*c^(3/2)*e^4*x^4 + 40*c^(3/2)*d*e^3*x^3 - 78*c^(3/2)*d^2*e^2*x^2 - 104*c^(3/2)*d^3*e*x + 107*c^(3/2)
*d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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mupad [B]  time = 0.58, size = 109, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {214\,c\,d^4\,\sqrt {d+e\,x}}{315\,e^2}-\frac {52\,c\,d^2\,x^2\,\sqrt {d+e\,x}}{105}+\frac {2\,c\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {208\,c\,d^3\,x\,\sqrt {d+e\,x}}{315\,e}+\frac {16\,c\,d\,e\,x^3\,\sqrt {d+e\,x}}{63}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((214*c*d^4*(d + e*x)^(1/2))/(315*e^2) - (52*c*d^2*x^2*(d + e*x)^(1/2))/105 + (2*c
*e^2*x^4*(d + e*x)^(1/2))/9 - (208*c*d^3*x*(d + e*x)^(1/2))/(315*e) + (16*c*d*e*x^3*(d + e*x)^(1/2))/63))/(x +
d/e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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