3.23.94 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac {247 \sqrt {5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac {13585 \sqrt {5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac {149435 \sqrt {5 x+3} \sqrt {1-2 x}}{448 (3 x+2)}-\frac {1643785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \]

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Rubi [A]  time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac {247 \sqrt {5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac {13585 \sqrt {5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac {149435 \sqrt {5 x+3} \sqrt {1-2 x}}{448 (3 x+2)}-\frac {1643785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^5*Sqrt[3 + 5*x]),x]

[Out]

(3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (247*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(168*(2 + 3*x)^3) + (
13585*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(672*(2 + 3*x)^2) + (149435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(448*(2 + 3*x))
- (1643785*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247}{56} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585}{336} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585 (1-2 x)^{3/2} \sqrt {3+5 x}}{672 (2+3 x)^2}+\frac {149435}{448} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585 (1-2 x)^{3/2} \sqrt {3+5 x}}{672 (2+3 x)^2}+\frac {149435 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {1643785}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585 (1-2 x)^{3/2} \sqrt {3+5 x}}{672 (2+3 x)^2}+\frac {149435 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {1643785}{448} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {3 (1-2 x)^{7/2} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {247 (1-2 x)^{5/2} \sqrt {3+5 x}}{168 (2+3 x)^3}+\frac {13585 (1-2 x)^{3/2} \sqrt {3+5 x}}{672 (2+3 x)^2}+\frac {149435 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}-\frac {1643785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 104, normalized size = 0.69 \begin {gather*} \frac {247 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (15707 x^2+21638 x+7488\right )}{(3 x+2)^3}-19965 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{9408}+\frac {3 \sqrt {5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^5*Sqrt[3 + 5*x]),x]

[Out]

(3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (247*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(7488 + 21638*x + 15
707*x^2))/(2 + 3*x)^3 - 19965*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/9408

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IntegrateAlgebraic [A]  time = 0.31, size = 122, normalized size = 0.81 \begin {gather*} \frac {1331 \sqrt {1-2 x} \left (\frac {9735 (1-2 x)^3}{(5 x+3)^3}+\frac {126217 (1-2 x)^2}{(5 x+3)^2}+\frac {665665 (1-2 x)}{5 x+3}+1270815\right )}{1344 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^4}-\frac {1643785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)^(5/2)/((2 + 3*x)^5*Sqrt[3 + 5*x]),x]

[Out]

(1331*Sqrt[1 - 2*x]*(1270815 + (9735*(1 - 2*x)^3)/(3 + 5*x)^3 + (126217*(1 - 2*x)^2)/(3 + 5*x)^2 + (665665*(1
- 2*x))/(3 + 5*x)))/(1344*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^4) - (1643785*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(448*Sqrt[7])

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fricas [A]  time = 0.90, size = 116, normalized size = 0.77 \begin {gather*} -\frac {4931355 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (11637735 \, x^{3} + 23794744 \, x^{2} + 16236916 \, x + 3699216\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/18816*(4931355*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(11637735*x^3 + 23794744*x^2 + 16236916*x + 3699216)*sqrt(5*x + 3)*sq
rt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [B]  time = 3.25, size = 368, normalized size = 2.44 \begin {gather*} \frac {328757}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {6655 \, \sqrt {10} {\left (1947 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1009736 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 213012800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {16266432000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {65065728000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{672 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

328757/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 6655/672*sqrt(10)*(1947*((sqrt(2)*sqrt(-10*x + 5
) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 1009736*((sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 213012800*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 +
16266432000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 65065728000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22)))^2 + 280)^4

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maple [B]  time = 0.02, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (399439755 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{18816 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)/(3*x+2)^5/(5*x+3)^(1/2),x)

[Out]

1/18816*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(399439755*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
[phone]*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^2*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+473410080*7^(1/2)*x*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+78901680*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+51789024*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3
*x+2)^4

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maxima [A]  time = 1.37, size = 143, normalized size = 0.95 \begin {gather*} \frac {1643785}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{36 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1477 \, \sqrt {-10 \, x^{2} - x + 3}}{216 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {37091 \, \sqrt {-10 \, x^{2} - x + 3}}{864 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {3879245 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

1643785/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/36*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) + 1477/216*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 37091/864*sqrt
(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 3879245/12096*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^5\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^5*(5*x + 3)^(1/2)),x)

[Out]

int((1 - 2*x)^(5/2)/((3*x + 2)^5*(5*x + 3)^(1/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)/(2+3*x)**5/(3+5*x)**(1/2),x)

[Out]

Timed out

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