3.2629 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{203 (3 x+2)^2}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-203*(2 + 3*x)^2)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))
+ (Sqrt[1 - 2*x]*(627287 + 991010*x))/(2196150*(3 + 5*x)^(3/2)) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sq
rt[10])

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Rubi [A]  time = 0.0305552, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 150, 145, 54, 216} \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{203 (3 x+2)^2}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(-203*(2 + 3*x)^2)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))
+ (Sqrt[1 - 2*x]*(627287 + 991010*x))/(2196150*(3 + 5*x)^(3/2)) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sq
rt[10])

Rule 98

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 150

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^2 \left (78+\frac{297 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{203 (2+3 x)^2}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-\frac{2049}{2}-\frac{9801 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{203 (2+3 x)^2}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac{81}{100} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{203 (2+3 x)^2}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac{81 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{50 \sqrt{5}}\\ &=-\frac{203 (2+3 x)^2}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x} (627287+991010 x)}{2196150 (3+5 x)^{3/2}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{50 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 0.211915, size = 208, normalized size = 1.84 \[ \frac{2401 \left (\frac{640 (1-2 x) (3 x+2)^4 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},2,2,2,\frac{7}{2}\right \},\left \{1,1,1,\frac{9}{2}\right \},\frac{5}{11} (1-2 x)\right )}{184877}+\frac{8000 \left (6 x^2+x-2\right )^3 \, _2F_1\left (\frac{3}{2},\frac{11}{2};\frac{13}{2};\frac{5}{11} (1-2 x)\right )}{5021863}+\frac{\sqrt{10-20 x} \sqrt{5 x+3} \left (104976000 x^7+31298400 x^6-23823180 x^5-179946603 x^4+114920076 x^3+695191648 x^2+1209328624 x+353337912\right )+43923 \left (19521 x^4-40932 x^3-387936 x^2-241968 x-62504\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15846600 \sqrt{55} (1-2 x)^{5/2}}\right )}{726 \sqrt{22} (1-2 x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2401*((Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*(353337912 + 1209328624*x + 695191648*x^2 + 114920076*x^3 - 179946603*x^
4 - 23823180*x^5 + 31298400*x^6 + 104976000*x^7) + 43923*(-62504 - 241968*x - 387936*x^2 - 40932*x^3 + 19521*x
^4)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(15846600*Sqrt[55]*(1 - 2*x)^(5/2)) + (8000*(-2 + x + 6*x^2)^3*Hypergeom
etric2F1[3/2, 11/2, 13/2, (5*(1 - 2*x))/11])/5021863 + (640*(1 - 2*x)*(2 + 3*x)^4*HypergeometricPFQ[{-1/2, 2,
2, 2, 7/2}, {1, 1, 1, 9/2}, (5*(1 - 2*x))/11])/184877))/(726*Sqrt[22]*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.013, size = 165, normalized size = 1.5 \begin{align*}{\frac{1}{43923000\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 355776300\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-209908017\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-21346578\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1026687660\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-60296260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/43923000*(1-2*x)^(1/2)*(355776300*10^(1/2)*arcsin(20/11*x+1/11)*x^4+71155260*10^(1/2)*arcsin(20/11*x+1/11)*x
^3-209908017*10^(1/2)*arcsin(20/11*x+1/11)*x^2+994040800*x^3*(-10*x^2-x+3)^(1/2)-21346578*10^(1/2)*arcsin(20/1
1*x+1/11)*x+1026687660*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3
)^(1/2)-60296260*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [B]  time = 2.8747, size = 243, normalized size = 2.15 \begin{align*} \frac{27}{1464100} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{81}{1000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{9747}{732050} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{1588351 \, x}{1098075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{108 \, x^{2}}{5 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{34823}{1098075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{86854 \, x}{9075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{12682}{9075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

27/1464100*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x + 3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) +
21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 81/1000*sqrt(10)*arcsin(-20/11*x - 1/11) +
9747/732050*sqrt(-10*x^2 - x + 3) - 1588351/1098075*x/sqrt(-10*x^2 - x + 3) + 108/5*x^2/(-10*x^2 - x + 3)^(3/2
) - 34823/1098075/sqrt(-10*x^2 - x + 3) + 86854/9075*x/(-10*x^2 - x + 3)^(3/2) - 12682/9075/(-10*x^2 - x + 3)^
(3/2)

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Fricas [A]  time = 1.80746, size = 369, normalized size = 3.27 \begin{align*} -\frac{3557763 \, \sqrt{10}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \,{\left (49702040 \, x^{3} + 51334383 \, x^{2} + 7883562 \, x - 3014813\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/43923000*(3557763*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(49702040*x^3 + 51334383*x^2 + 7883562*x - 3014813)*sqrt(5*x + 3)*sq
rt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.24857, size = 247, normalized size = 2.19 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{87846000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{81}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{26620 \, \sqrt{5 \, x + 3}} + \frac{343 \,{\left (232 \, \sqrt{5}{\left (5 \, x + 3\right )} - 891 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2196150 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{825 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{5490375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/87846000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 81/500*sqrt(10)*arcsin(1/11*sqrt
(22)*sqrt(5*x + 3)) - 1/26620*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 343/2196150*(232*s
qrt(5)*(5*x + 3) - 891*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/5490375*(825*sqrt(10)*(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3