3.8.28 \(\int \frac {(d+e x)^{7/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\)

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

________________________________________________________________________________________

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 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(64*d^2*Sqrt[d + e*x])/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (16*d*(d + e*x)^(3/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]
) - (2*(d + e*x)^(5/2))/(3*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)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} (8 d) \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} \left (32 d^2\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 55, normalized size = 0.46 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-23*d^2 + 10*d*e*x + e^2*x^2))/(3*c*e*Sqrt[c*(d^2 - e^2*x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.04, size = 73, normalized size = 0.61 \begin {gather*} \frac {2 \left (-32 d^2+8 d (d+e x)+(d+e x)^2\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{3 c^2 e (e x-d) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(2*(-32*d^2 + 8*d*(d + e*x) + (d + e*x)^2)*Sqrt[2*c*d*(d + e*x) - c*(d + e*x)^2])/(3*c^2*e*(-d + e*x)*Sqrt[d +
e*x])

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 66, normalized size = 0.55 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e^{2} x^{2} + 10 \, d e x - 23 \, d^{2}\right )} \sqrt {e x + d}}{3 \, {\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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-c*e^2*x^2 + c*d^2)*(e^2*x^2 + 10*d*e*x - 23*d^2)*sqrt(e*x + d)/(c^2*e^3*x^2 - c^2*d^2*e)

________________________________________________________________________________________

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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 55, normalized size = 0.46 \begin {gather*} \frac {2 \left (-e x +d \right ) \left (-e^{2} x^{2}-10 d x e +23 d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/3*(-e*x+d)*(-e^2*x^2-10*d*e*x+23*d^2)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.52, size = 43, normalized size = 0.36 \begin {gather*} -\frac {2 \, {\left (\sqrt {c} e^{2} x^{2} + 10 \, \sqrt {c} d e x - 23 \, \sqrt {c} d^{2}\right )}}{3 \, \sqrt {-e x + d} c^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(sqrt(c)*e^2*x^2 + 10*sqrt(c)*d*e*x - 23*sqrt(c)*d^2)/(sqrt(-e*x + d)*c^2*e)

________________________________________________________________________________________

mupad [B]  time = 0.67, size = 86, normalized size = 0.72 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,e}-\frac {46\,d^2\,\sqrt {d+e\,x}}{3\,c^2\,e^3}+\frac {20\,d\,x\,\sqrt {d+e\,x}}{3\,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)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((2*x^2*(d + e*x)^(1/2))/(3*c^2*e) - (46*d^2*(d + e*x)^(1/2))/(3*c^2*e^3) + (20*d*x
*(d + e*x)^(1/2))/(3*c^2*e^2)))/(x^2 - d^2/e^2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {7}{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)**(7/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

________________________________________________________________________________________