3.29.29 \(\int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\) [2829]

Optimal. Leaf size=191 \[ \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 \sqrt {2+3 x}}-\frac {184636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}-\frac {9124 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105} \]

[Out]

-184636/756315*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-9124/756315*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/147*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-536/5145*(1-2*x)^(1
/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+974/36015*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+184636/252105*(1-2*x)^(1/2
)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 191, 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 = {100, 157, 164, 114, 120} \begin {gather*} -\frac {9124 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}-\frac {184636 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}+\frac {184636 \sqrt {1-2 x} \sqrt {5 x+3}}{252105 \sqrt {3 x+2}}+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{36015 (3 x+2)^{3/2}}-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{5145 (3 x+2)^{5/2}}+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)),x]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) - (536*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*(2 + 3*x)^(5/2
)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 + 3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*
Sqrt[2 + 3*x]) - (184636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105 - (9124*Sqrt[11/
3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/252105

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {2}{147} \int \frac {-347-\frac {1175 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}-\frac {4 \int \frac {-\frac {5847}{4}-2010 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {41289}{4}+\frac {7305 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{108045}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {906135}{8}-\frac {692385 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{756315}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 \sqrt {2+3 x}}+\frac {50182 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{252105}+\frac {184636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{252105}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 \sqrt {2+3 x}}-\frac {184636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}-\frac {9124 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}\\ \end {align*}

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Mathematica [A]
time = 4.71, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (727631+3324960 x+5015853 x^2+2492586 x^3\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (92318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-17045 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{756315} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(727631 + 3324960*x + 5015853*x^2 + 2492586*x^3))/(2 + 3*x)^(7/2) + Sqrt[2]
*(92318*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 17045*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])))/756315

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).
time = 0.10, size = 401, normalized size = 2.10

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {536 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{138915 \left (\frac {2}{3}+x \right )^{3}}+\frac {974 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{324135 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {369272}{50421} x^{2}-\frac {184636}{252105} x +\frac {184636}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {120818 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1058841 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {184636 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1058841 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11907 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(273\)
default	\(-\frac {2 \left (2032371 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2492586 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4064742 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4985172 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2709828 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3323448 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+602184 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-738544 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-74777580 x^{5}-157953348 x^{4}-92363085 x^{3}[phone] x^{2}[phone] x +6548679\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{756315 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/756315*(2032371*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x
)^(1/2)-2492586*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)+4064742*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2
)-4985172*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+2
709828*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-332344
8*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+602184*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))-738544*2^(1/2)*(2+3
*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-74777580*x^5-157953348*x^4-
92363085*x^3+13338867*x^2+27741747*x+6548679)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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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((3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)/((3*x + 2)^(9/2)*sqrt(-2*x + 1)), x)

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Fricas [A]
time = 0.33, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (2492586 \, x^{3} + 5015853 \, x^{2} + 3324960 \, x + 727631\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{252105 \, {\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((3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

2/252105*(2492586*x^3 + 5015853*x^2 + 3324960*x + 727631)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16)

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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((3+5*x)**(3/2)/(2+3*x)**(9/2)/(1-2*x)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4847 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((3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)/((3*x + 2)^(9/2)*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 (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(3/2)/((1 - 2*x)^(1/2)*(3*x + 2)^(9/2)),x)

[Out]

int((5*x + 3)^(3/2)/((1 - 2*x)^(1/2)*(3*x + 2)^(9/2)), x)

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