3.12.1 \(\int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx\) [1101]

Optimal. Leaf size=81 \[ \frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{3003 (1-x)^{7/2}} \]

[Out]

1/13*(1+x)^(7/2)/(1-x)^(13/2)+3/143*(1+x)^(7/2)/(1-x)^(11/2)+2/429*(1+x)^(7/2)/(1-x)^(9/2)+2/3003*(1+x)^(7/2)/
(1-x)^(7/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \begin {gather*} \frac {2 (x+1)^{7/2}}{3003 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{429 (1-x)^{9/2}}+\frac {3 (x+1)^{7/2}}{143 (1-x)^{11/2}}+\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(15/2),x]

[Out]

(1 + x)^(7/2)/(13*(1 - x)^(13/2)) + (3*(1 + x)^(7/2))/(143*(1 - x)^(11/2)) + (2*(1 + x)^(7/2))/(429*(1 - x)^(9
/2)) + (2*(1 + x)^(7/2))/(3003*(1 - x)^(7/2))

Rule 37

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx &=\frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3}{13} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {6}{143} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2}{429} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac {3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{3003 (1-x)^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 35, normalized size = 0.43 \begin {gather*} \frac {(1+x)^{7/2} \left (310-97 x+20 x^2-2 x^3\right )}{3003 (1-x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(15/2),x]

[Out]

((1 + x)^(7/2)*(310 - 97*x + 20*x^2 - 2*x^3))/(3003*(1 - x)^(13/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(57)=114\).
time = 0.16, size = 128, normalized size = 1.58

method	result	size
gosper	\(-\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{3}-20 x^{2}+97 x -310\right )}{3003 \left (1-x \right )^{\frac {13}{2}}}\)	\(30\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{7}-12 x^{6}+29 x^{5}-34 x^{4}-736 x^{3}-1492 x^{2}-1143 x -310\right )}{3003 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{6} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(76\)
default	\(\frac {\left (1+x \right )^{\frac {5}{2}}}{4 \left (1-x \right )^{\frac {13}{2}}}-\frac {\left (1+x \right )^{\frac {3}{2}}}{4 \left (1-x \right )^{\frac {13}{2}}}+\frac {3 \sqrt {1+x}}{26 \left (1-x \right )^{\frac {13}{2}}}-\frac {3 \sqrt {1+x}}{572 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{1716 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{1001 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{3003 \sqrt {1-x}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(15/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(1+x)^(5/2)/(1-x)^(13/2)-1/4*(1+x)^(3/2)/(1-x)^(13/2)+3/26*(1+x)^(1/2)/(1-x)^(13/2)-3/572*(1+x)^(1/2)/(1-x
)^(11/2)-5/1716*(1+x)^(1/2)/(1-x)^(9/2)-5/3003*(1+x)^(1/2)/(1-x)^(7/2)-1/1001*(1+x)^(1/2)/(1-x)^(5/2)-2/3003*(
1+x)^(1/2)/(1-x)^(3/2)-2/3003*(1+x)^(1/2)/(1-x)^(1/2)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (57) = 114\).
time = 0.27, size = 325, normalized size = 4.01 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{4 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} - \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{4 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{26 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{572 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{1716 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1001 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(15/2),x, algorithm="maxima")

[Out]

-1/4*(-x^2 + 1)^(5/2)/(x^9 - 9*x^8 + 36*x^7 - 84*x^6 + 126*x^5 - 126*x^4 + 84*x^3 - 36*x^2 + 9*x - 1) - 1/4*(-
x^2 + 1)^(3/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) - 3/26*sqrt(-x^2 + 1)/(x^7
- 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 3/572*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3
+ 15*x^2 - 6*x + 1) + 5/1716*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 5/3003*sqrt(-x^2 + 1)/
(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/1001*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/3003*sqrt(-x^2 + 1)/(x^2 -
2*x + 1) + 2/3003*sqrt(-x^2 + 1)/(x - 1)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (57) = 114\).
time = 1.35, size = 115, normalized size = 1.42 \begin {gather*} \frac {310 \, x^{7} - 2170 \, x^{6} + 6510 \, x^{5} - 10850 \, x^{4} + 10850 \, x^{3} - 6510 \, x^{2} + {\left (2 \, x^{6} - 14 \, x^{5} + 43 \, x^{4} - 77 \, x^{3} - 659 \, x^{2} - 833 \, x - 310\right )} \sqrt {x + 1} \sqrt {-x + 1} + 2170 \, x - 310}{3003 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(15/2),x, algorithm="fricas")

[Out]

1/3003*(310*x^7 - 2170*x^6 + 6510*x^5 - 10850*x^4 + 10850*x^3 - 6510*x^2 + (2*x^6 - 14*x^5 + 43*x^4 - 77*x^3 -
659*x^2 - 833*x - 310)*sqrt(x + 1)*sqrt(-x + 1) + 2170*x - 310)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*
x^2 + 7*x - 1)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(15/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4061 deep

________________________________________________________________________________________

Giac [A]
time = 1.96, size = 35, normalized size = 0.43 \begin {gather*} \frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 12\right )} + 143\right )} {\left (x + 1\right )} - 429\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{3003 \, {\left (x - 1\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(15/2),x, algorithm="giac")

[Out]

1/3003*((2*(x + 1)*(x - 12) + 143)*(x + 1) - 429)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^7

________________________________________________________________________________________

Mupad [B]
time = 0.31, size = 110, normalized size = 1.36 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {119\,x\,\sqrt {x+1}}{429}+\frac {310\,\sqrt {x+1}}{3003}+\frac {659\,x^2\,\sqrt {x+1}}{3003}+\frac {x^3\,\sqrt {x+1}}{39}-\frac {43\,x^4\,\sqrt {x+1}}{3003}+\frac {2\,x^5\,\sqrt {x+1}}{429}-\frac {2\,x^6\,\sqrt {x+1}}{3003}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(5/2)/(1 - x)^(15/2),x)

[Out]

-((1 - x)^(1/2)*((119*x*(x + 1)^(1/2))/429 + (310*(x + 1)^(1/2))/3003 + (659*x^2*(x + 1)^(1/2))/3003 + (x^3*(x
+ 1)^(1/2))/39 - (43*x^4*(x + 1)^(1/2))/3003 + (2*x^5*(x + 1)^(1/2))/429 - (2*x^6*(x + 1)^(1/2))/3003))/(7*x
- 21*x^2 + 35*x^3 - 35*x^4 + 21*x^5 - 7*x^6 + x^7 - 1)

________________________________________________________________________________________