3.951 \(\int (d+e x)^m (d^2-e^2 x^2)^{7/2} \, dx\)

Optimal. Leaf size=59 \[ \frac {\left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \, _2F_1\left (1,m+9;m+\frac {11}{2};\frac {d+e x}{2 d}\right )}{d e (2 m+9)} \]

[Out]

(e*x+d)^m*(-e^2*x^2+d^2)^(9/2)*hypergeom([1, 9+m],[11/2+m],1/2*(e*x+d)/d)/d/e/(9+2*m)

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Rubi [A]  time = 0.05, 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 = {680, 678, 69} \[ -\frac {2^{m+\frac {9}{2}} \left (d^2-e^2 x^2\right )^{9/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {9}{2}} \, _2F_1\left (\frac {9}{2},-m-\frac {7}{2};\frac {11}{2};\frac {d-e x}{2 d}\right )}{9 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2^(9/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-9/2 - m)*(d^2 - e^2*x^2)^(9/2)*Hypergeometric2F1[9/2, -7/2 - m, 11/2
, (d - e*x)/(2*d)])/(9*d*e)

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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 678

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*x)/e)^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 680

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 )^{7/2} \, 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 )^{7/2} \, dx\\ &=\frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {9}{2}-m} \left (d^2-e^2 x^2\right )^{9/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {7}{2}+m} \left (d^2-d e x\right )^{7/2} \, dx}{\left (d^2-d e x\right )^{9/2}}\\ &=-\frac {2^{\frac {9}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {9}{2}-m} \left (d^2-e^2 x^2\right )^{9/2} \, _2F_1\left (\frac {9}{2},-\frac {7}{2}-m;\frac {11}{2};\frac {d-e x}{2 d}\right )}{9 d e}\\ \end {align*}

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Mathematica [C]  time = 0.49, size = 347, normalized size = 5.88 \[ \frac {(d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {1}{2}} \left (-105 d^4 e^3 x^3 \sqrt {d-e x} \sqrt {d+e x} F_1\left (3;-\frac {1}{2},-m-\frac {1}{2};4;\frac {e x}{d},-\frac {e x}{d}\right )+63 d^2 e^5 x^5 \sqrt {d-e x} \sqrt {d+e x} F_1\left (5;-\frac {1}{2},-m-\frac {1}{2};6;\frac {e x}{d},-\frac {e x}{d}\right )-15 e^7 x^7 \sqrt {d-e x} \sqrt {d+e x} F_1\left (7;-\frac {1}{2},-m-\frac {1}{2};8;\frac {e x}{d},-\frac {e x}{d}\right )-35 d^7 2^{m+\frac {3}{2}} \sqrt {1-\frac {e x}{d}} \sqrt {d^2-e^2 x^2} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {d-e x}{2 d}\right )+35 d^6 e 2^{m+\frac {3}{2}} x \sqrt {1-\frac {e x}{d}} \sqrt {d^2-e^2 x^2} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {d-e x}{2 d}\right )\right )}{105 e \sqrt {1-\frac {e x}{d}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*(-105*d^4*e^3*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[3, -1/2, -1/2 - m
, 4, (e*x)/d, -((e*x)/d)] + 63*d^2*e^5*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[5, -1/2, -1/2 - m, 6, (e*x)/d,
-((e*x)/d)] - 15*e^7*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]*AppellF1[7, -1/2, -1/2 - m, 8, (e*x)/d, -((e*x)/d)] - 35
*2^(3/2 + m)*d^7*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)]
+ 35*2^(3/2 + m)*d^6*e*x*Sqrt[1 - (e*x)/d]*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)
/(2*d)]))/(105*e*Sqrt[1 - (e*x)/d])

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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (e^{6} x^{6} - 3 \, d^{2} e^{4} x^{4} + 3 \, d^{4} e^{2} x^{2} - d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

integral(-(e^6*x^6 - 3*d^2*e^4*x^4 + 3*d^4*e^2*x^2 - d^6)*sqrt(-e^2*x^2 + d^2)*(e*x + d)^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^m, x)

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maple [F]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} \left (e x +d \right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(-e^2*x^2+d^2)^(7/2),x)

[Out]

int((e*x+d)^m*(-e^2*x^2+d^2)^(7/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^m,x)

[Out]

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^m, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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