3.29.59 \(\int \frac {(2+3 x)^{7/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\) [2859]

Optimal. Leaf size=158 \[ -\frac {15553 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{8750}-\frac {333}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {270248 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{21875}-\frac {178879 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750 \sqrt {33}} \]

[Out]

-270248/65625*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-178879/1443750*EllipticF(1/7*21^(
1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-333/875*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-3/35*(2+3*x)^(5
/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-15553/8750*(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 = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {104, 159, 164, 114, 120} \begin {gather*} -\frac {178879 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750 \sqrt {33}}-\frac {270248 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{21875}-\frac {3}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {333}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {15553 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{8750} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(7/2)/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

(-15553*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/8750 - (333*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/87
5 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/35 - (270248*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/21875 - (178879*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(43750*Sqrt[33])

Rule 104

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

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)^{7/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {1}{35} \int \frac {\left (-\frac {409}{2}-333 x\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {333}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {1}{875} \int \frac {\sqrt {2+3 x} \left (\frac {28775}{2}+\frac {46659 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {15553 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{8750}-\frac {333}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {\int \frac {-\frac {2053113}{4}-810744 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{13125}\\ &=-\frac {15553 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{8750}-\frac {333}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {178879 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{87500}+\frac {270248 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{21875}\\ &=-\frac {15553 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{8750}-\frac {333}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {3}{35} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {270248 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{21875}-\frac {178879 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{43750 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 3.86, size = 97, normalized size = 0.61 \begin {gather*} \frac {-15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (25213+18990 x+6750 x^2\right )+1080992 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-544355 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{131250 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(7/2)/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

(-15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(25213 + 18990*x + 6750*x^2) + 1080992*EllipticE[ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]], -33/2] - 544355*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(131250*Sqrt[2])

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Maple [A]
time = 0.10, size = 148, normalized size = 0.94

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (536637 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1080992 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+6075000 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-8712930 x -4538340\right )}{262500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(148\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {27 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35}-\frac {1899 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}-\frac {25213 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8750}+\frac {684371 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{367500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {270248 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{91875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
risch	\(\frac {\left (6750 x^{2}+18990 x +25213\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{8750 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {684371 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{962500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {24568 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{21875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/262500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(536637*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)
*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-1080992*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+6075000*x^5+21748500*x^4+34377300*x^3+12194070*x^2-8712930*x-4538340)/(3
0*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)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)/(sqrt(5*x + 3)*sqrt(-2*x + 1)), x)

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Fricas [A]
time = 0.28, size = 33, normalized size = 0.21 \begin {gather*} -\frac {1}{8750} \, {\left (6750 \, x^{2} + 18990 \, x + 25213\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/8750*(6750*x^2 + 18990*x + 25213)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)

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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((2+3*x)**(7/2)/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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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)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)/(sqrt(5*x + 3)*sqrt(-2*x + 1)), 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 )}^{7/2}}{\sqrt {1-2\,x}\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^(7/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(1/2)),x)

[Out]

int((3*x + 2)^(7/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(1/2)), x)

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