3.862 \(\int (d+e x)^{5/2} \sqrt {c d^2-c e^2 x^2} \, dx\)

Optimal. Leaf size=160 \[ -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt {d+e x}}-\frac {8 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}-\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{3/2}}{315 c e (d+e x)^{3/2}} \]

[Out]

-256/315*d^3*(-c*e^2*x^2+c*d^2)^(3/2)/c/e/(e*x+d)^(3/2)-2/9*(e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(3/2)/c/e-64/105*
d^2*(-c*e^2*x^2+c*d^2)^(3/2)/c/e/(e*x+d)^(1/2)-8/21*d*(-c*e^2*x^2+c*d^2)^(3/2)*(e*x+d)^(1/2)/c/e

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Rubi [A]  time = 0.07, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{3/2}}{315 c e (d+e x)^{3/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt {d+e x}}-\frac {8 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(3/2))/(315*c*e*(d + e*x)^(3/2)) - (64*d^2*(c*d^2 - c*e^2*x^2)^(3/2))/(105*c*e*S
qrt[d + e*x]) - (8*d*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2))/(21*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2
)^(3/2))/(9*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 (d+e x)^{5/2} \sqrt {c d^2-c e^2 x^2} \, dx &=-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}+\frac {1}{3} (4 d) \int (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2} \, dx\\ &=-\frac {8 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}+\frac {1}{21} \left (32 d^2\right ) \int \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2} \, dx\\ &=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt {d+e x}}-\frac {8 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}+\frac {1}{105} \left (128 d^3\right ) \int \frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{3/2}}{315 c e (d+e x)^{3/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt {d+e x}}-\frac {8 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 75, normalized size = 0.47 \[ -\frac {2 \left (319 d^4+2 d^3 e x-156 d^2 e^2 x^2-130 d e^3 x^3-35 e^4 x^4\right ) \sqrt {c \left (d^2-e^2 x^2\right )}}{315 e \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c*(d^2 - e^2*x^2)]*(319*d^4 + 2*d^3*e*x - 156*d^2*e^2*x^2 - 130*d*e^3*x^3 - 35*e^4*x^4))/(315*e*Sqrt[
d + e*x])

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fricas [A]  time = 0.63, size = 78, normalized size = 0.49 \[ \frac {2 \, {\left (35 \, e^{4} x^{4} + 130 \, d e^{3} x^{3} + 156 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 319 \, d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*e^4*x^4 + 130*d*e^3*x^3 + 156*d^2*e^2*x^2 - 2*d^3*e*x - 319*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 \[ \int \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + d\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 66, normalized size = 0.41 \[ -\frac {2 \left (-e x +d \right ) \left (35 e^{3} x^{3}+165 e^{2} x^{2} d +321 d^{2} x e +319 d^{3}\right ) \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{315 \sqrt {e x +d}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(-e*x+d)*(35*e^3*x^3+165*d*e^2*x^2+321*d^2*e*x+319*d^3)*(-c*e^2*x^2+c*d^2)^(1/2)/e/(e*x+d)^(1/2)

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maxima [A]  time = 1.47, size = 82, normalized size = 0.51 \[ \frac {2 \, {\left (35 \, \sqrt {c} e^{4} x^{4} + 130 \, \sqrt {c} d e^{3} x^{3} + 156 \, \sqrt {c} d^{2} e^{2} x^{2} - 2 \, \sqrt {c} d^{3} e x - 319 \, \sqrt {c} d^{4}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*sqrt(c)*e^4*x^4 + 130*sqrt(c)*d*e^3*x^3 + 156*sqrt(c)*d^2*e^2*x^2 - 2*sqrt(c)*d^3*e*x - 319*sqrt(c)*
d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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mupad [B]  time = 0.63, size = 103, normalized size = 0.64 \[ \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {104\,d^2\,x^2\,\sqrt {d+e\,x}}{105}-\frac {638\,d^4\,\sqrt {d+e\,x}}{315\,e^2}+\frac {2\,e^2\,x^4\,\sqrt {d+e\,x}}{9}+\frac {52\,d\,e\,x^3\,\sqrt {d+e\,x}}{63}-\frac {4\,d^3\,x\,\sqrt {d+e\,x}}{315\,e}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((104*d^2*x^2*(d + e*x)^(1/2))/105 - (638*d^4*(d + e*x)^(1/2))/(315*e^2) + (2*e^2*x
^4*(d + e*x)^(1/2))/9 + (52*d*e*x^3*(d + e*x)^(1/2))/63 - (4*d^3*x*(d + e*x)^(1/2))/(315*e)))/(x + d/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(-c*(-d + e*x)*(d + e*x))*(d + e*x)**(5/2), x)

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