3.10.53 \(\int (d+e x)^m (d^2-e^2 x^2)^{3/2} \, dx\) [953]

Optimal. Leaf size=59 \[ \frac {(d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (1,5+m;\frac {7}{2}+m;\frac {d+e x}{2 d}\right )}{d e (5+2 m)} \]

[Out]

(e*x+d)^m*(-e^2*x^2+d^2)^(5/2)*hypergeom([1, 5+m],[7/2+m],1/2*(e*x+d)/d)/d/e/(5+2*m)

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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {694, 692, 71} \begin {gather*} -\frac {2^{m+\frac {5}{2}} \left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {5}{2}} \, _2F_1\left (\frac {5}{2},-m-\frac {3}{2};\frac {7}{2};\frac {d-e x}{2 d}\right )}{5 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(2^(5/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-5/2 - m)*(d^2 - e^2*x^2)^(5/2)*Hypergeometric2F1[5/2, -3/2 - m,
7/2, (d - e*x)/(2*d)])/(d*e)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(m - 1)*((a + c*x^2)^(p + 1)/((1
+ e*(x/d))^(p + 1)*(a/d + (c*x)/e)^(p + 1))), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))

Rule 694

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

Rubi steps

\begin {align*} \int (d+e x)^m \left (d^2-e^2 x^2\right )^{3/2} \, dx &=\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \left (1+\frac {e x}{d}\right )^m \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {3}{2}+m} \left (d^2-d e x\right )^{3/2} \, dx}{\left (d^2-d e x\right )^{5/2}}\\ &=-\frac {2^{\frac {5}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (\frac {5}{2},-\frac {3}{2}-m;\frac {7}{2};\frac {d-e x}{2 d}\right )}{5 d e}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.48, size = 191, normalized size = 3.24 \begin {gather*} -\frac {2^m (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-2 m} \left (e^3 x^3 \sqrt {d-e x} \sqrt {d+e x} \left (\frac {1}{2}+\frac {e x}{2 d}\right )^m F_1\left (3;-\frac {1}{2},-\frac {1}{2}-m;4;\frac {e x}{d},-\frac {e x}{d}\right )+2 d^2 (d-e x) \sqrt {2-\frac {2 e x}{d}} \left (1+\frac {e x}{d}\right )^m \sqrt {d^2-e^2 x^2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {d-e x}{2 d}\right )\right )}{3 e \sqrt {1-\frac {e x}{d}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*(2^m*(d + e*x)^m*(1 + (e*x)/d)^(-1/2 - 2*m)*(e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*(1/2 + (e*x)/(2*d))^m*Ap
pellF1[3, -1/2, -1/2 - m, 4, (e*x)/d, -((e*x)/d)] + 2*d^2*(d - e*x)*Sqrt[2 - (2*e*x)/d]*(1 + (e*x)/d)^m*Sqrt[d
^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)]))/(e*Sqrt[1 - (e*x)/d])

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(x^2*e^2 - d^2)*sqrt(-x^2*e^2 + d^2)*(x*e + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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