3.29.100 \(\int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\) [2900]

Optimal. Leaf size=188 \[ \frac {9694}{175} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1289089 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}+\frac {9694}{875} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

1289089/10500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+9694/2625*EllipticF(1/7*21^(1/2)*
(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+(2+3*x)^(5/2)*(3+5*x)^(3/2)/(1-2*x)^(1/2)+12/7*(2+3*x)^(3/2)*(3+5*x)^(
3/2)*(1-2*x)^(1/2)+2511/350*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+9694/175*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+
5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \begin {gather*} \frac {9694}{875} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {1289089 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}+\frac {(5 x+3)^{3/2} (3 x+2)^{5/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac {2511}{350} \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}+\frac {9694}{175} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9694*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/175 + (2511*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/350
+ (12*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/7 + ((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (12
89089*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 + (9694*Sqrt[11/3]*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/875

Rule 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{3/2} \sqrt {3+5 x} \left (\frac {75}{2}+60 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{35} \int \frac {\left (-3975-\frac {12555 x}{2}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {1}{875} \int \frac {\sqrt {3+5 x} \left (\frac {1133985}{4}+436230 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {9694}{175} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {\int \frac {-\frac {36724815}{4}-\frac {58009005 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7875}\\ &=\frac {9694}{175} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {53317}{875} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {1289089 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3500}\\ &=\frac {9694}{175} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2511}{350} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1289089 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}+\frac {9694}{875} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

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Mathematica [A]
time = 7.71, size = 115, normalized size = 0.61 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-34721+17487 x+8460 x^2+2250 x^3\right )-1289089 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+649285 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{10500 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-34721 + 17487*x + 8460*x^2 + 2250*x^3) - 1289089*Sqrt[2 - 4*x]*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 649285*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2])/(10500*Sqrt[1 - 2*x])

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Maple [A]
time = 0.10, size = 148, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (639804 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1289089 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1012500 x^{5}+5089500 x^{4}[phone] x^{3}-4134060 x^{2}-16643310 x -6249780\right )}{315000 x^{3}+241500 x^{2}-73500 x -63000}\)	\(148\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {45 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}+\frac {1917 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{140}+\frac {44559 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1400}-\frac {816107 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{14700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1289089 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {539 \left (-30 x^{2}-38 x -12\right )}{16 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(639804*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*E
llipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-1289089*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
E(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+1012500*x^5+5089500*x^4+13096350*x^3-4134060*x^2-16643310*x-6249780)/(30*x
^3+23*x^2-7*x-6)

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

[Out]

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

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Fricas [A]
time = 0.22, size = 45, normalized size = 0.24 \begin {gather*} \frac {{\left (2250 \, x^{3} + 8460 \, x^{2} + 17487 \, x - 34721\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{350 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/350*(2250*x^3 + 8460*x^2 + 17487*x - 34721)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [F(-1)] Timed out
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((2+3*x)**(5/2)*(3+5*x)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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