3.8.27 \(\int \frac {(d+e x)^{9/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}} \]

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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} \begin {gather*} -\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(256*d^3*Sqrt[d + e*x])/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (64*d^2*(d + e*x)^(3/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2]) - (8*d*(d + e*x)^(5/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(7/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2])

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)^{9/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} (12 d) \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} \left (32 d^2\right ) \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} \left (128 d^3\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 66, normalized size = 0.41 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-91 d^3+43 d^2 e x+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-91*d^3 + 43*d^2*e*x + 7*d*e^2*x^2 + e^3*x^3))/(5*c*e*Sqrt[c*(d^2 - e^2*x^2)])

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IntegrateAlgebraic [A]  time = 6.55, size = 85, normalized size = 0.53 \begin {gather*} \frac {2 \left (-128 d^3+32 d^2 (d+e x)+4 d (d+e x)^2+(d+e x)^3\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{5 c^2 e (e x-d) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(9/2)/(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*d^3 + 32*d^2*(d + e*x) + 4*d*(d + e*x)^2 + (d + e*x)^3))/(5*c^2
*e*(-d + e*x)*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 77, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{5 \, {\left (c^{2} e^{3} x^{2} - c^{2} d^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/5*(e^3*x^3 + 7*d*e^2*x^2 + 43*d^2*e*x - 91*d^3)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(c^2*e^3*x^2 - c^2*d^
2*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.04, size = 66, normalized size = 0.41 \begin {gather*} \frac {2 \left (-e x +d \right ) \left (-e^{3} x^{3}-7 e^{2} x^{2} d -43 d^{2} x e +91 d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/5*(-e*x+d)*(-e^3*x^3-7*d*e^2*x^2-43*d^2*e*x+91*d^3)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

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maxima [A]  time = 1.54, size = 45, normalized size = 0.28 \begin {gather*} -\frac {2 \, {\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )}}{5 \, \sqrt {-e x + d} c^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(e^3*x^3 + 7*d*e^2*x^2 + 43*d^2*e*x - 91*d^3)/(sqrt(-e*x + d)*c^(3/2)*e)

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mupad [B]  time = 0.75, size = 104, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}}{5\,c^2}-\frac {182\,d^3\,\sqrt {d+e\,x}}{5\,c^2\,e^3}+\frac {14\,d\,x^2\,\sqrt {d+e\,x}}{5\,c^2\,e}+\frac {86\,d^2\,x\,\sqrt {d+e\,x}}{5\,c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((2*x^3*(d + e*x)^(1/2))/(5*c^2) - (182*d^3*(d + e*x)^(1/2))/(5*c^2*e^3) + (14*d*x^
2*(d + e*x)^(1/2))/(5*c^2*e) + (86*d^2*x*(d + e*x)^(1/2))/(5*c^2*e^2)))/(x^2 - d^2/e^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 )^{\frac {9}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(9/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

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