3.502 \(\int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=225 \[ \frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-5 a B)}{768 b^2}-\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-5 a B)}{512 b^3}+\frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-5 a B)}{192 b}+\frac{a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac{x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b} \]

[Out]

-(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) + (a^3*(12*A*b - 5*a*B)*x^(3/2)*Sqrt[a + b*x])/(768*b^
2) + (a^2*(12*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/(192*b) + (a*(12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(3/2))/(96*b
) + ((12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(5/2))/(60*b) + (B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + (a^5*(12*A*b - 5*a
*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

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Rubi [A]  time = 0.105212, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {80, 50, 63, 217, 206} \[ \frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-5 a B)}{768 b^2}-\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-5 a B)}{512 b^3}+\frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-5 a B)}{192 b}+\frac{a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac{x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

-(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) + (a^3*(12*A*b - 5*a*B)*x^(3/2)*Sqrt[a + b*x])/(768*b^
2) + (a^2*(12*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/(192*b) + (a*(12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(3/2))/(96*b
) + ((12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(5/2))/(60*b) + (B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + (a^5*(12*A*b - 5*a
*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

Rule 80

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

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

Rule 206

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

Rubi steps

\begin{align*} \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx &=\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (6 A b-\frac{5 a B}{2}\right ) \int x^{3/2} (a+b x)^{5/2} \, dx}{6 b}\\ &=\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{(a (12 A b-5 a B)) \int x^{3/2} (a+b x)^{3/2} \, dx}{24 b}\\ &=\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (a^2 (12 A b-5 a B)\right ) \int x^{3/2} \sqrt{a+b x} \, dx}{64 b}\\ &=\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (a^3 (12 A b-5 a B)\right ) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{384 b}\\ &=\frac{a^3 (12 A b-5 a B) x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}-\frac{\left (a^4 (12 A b-5 a B)\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{512 b^2}\\ &=-\frac{a^4 (12 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{512 b^3}+\frac{a^3 (12 A b-5 a B) x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (a^5 (12 A b-5 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{1024 b^3}\\ &=-\frac{a^4 (12 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{512 b^3}+\frac{a^3 (12 A b-5 a B) x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (a^5 (12 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{512 b^3}\\ &=-\frac{a^4 (12 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{512 b^3}+\frac{a^3 (12 A b-5 a B) x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{\left (a^5 (12 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^3}\\ &=-\frac{a^4 (12 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{512 b^3}+\frac{a^3 (12 A b-5 a B) x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^2 (12 A b-5 a B) x^{5/2} \sqrt{a+b x}}{192 b}+\frac{a (12 A b-5 a B) x^{5/2} (a+b x)^{3/2}}{96 b}+\frac{(12 A b-5 a B) x^{5/2} (a+b x)^{5/2}}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.316186, size = 151, normalized size = 0.67 \[ \frac{\sqrt{a+b x} (12 A b-5 a B) \left (\sqrt{b} \sqrt{x} \sqrt{\frac{b x}{a}+1} \left (248 a^2 b^2 x^2+10 a^3 b x-15 a^4+336 a b^3 x^3+128 b^4 x^4\right )+15 a^{9/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{7680 b^{7/2} \sqrt{\frac{b x}{a}+1}}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + ((12*A*b - 5*a*B)*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[1 + (b*x)/a]*(-15*a^
4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 336*a*b^3*x^3 + 128*b^4*x^4) + 15*a^(9/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]
]))/(7680*b^(7/2)*Sqrt[1 + (b*x)/a])

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Maple [A]  time = 0.011, size = 302, normalized size = 1.3 \begin{align*}{\frac{1}{15360}\sqrt{x}\sqrt{bx+a} \left ( 2560\,B{x}^{5}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }+3072\,A{x}^{4}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }+6400\,B{x}^{4}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+8064\,A{x}^{3}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+4320\,B{x}^{3}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+5952\,A{x}^{2}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+80\,B{x}^{2}{a}^{3}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+240\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x{a}^{3}-100\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{4}+180\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{5}b-360\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{4}-75\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{6}+150\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{5} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

1/15360*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(2560*B*x^5*b^(11/2)*(x*(b*x+a))^(1/2)+3072*A*x^4*b^(11/2)*(x*(b*x+a))^(
1/2)+6400*B*x^4*a*b^(9/2)*(x*(b*x+a))^(1/2)+8064*A*x^3*a*b^(9/2)*(x*(b*x+a))^(1/2)+4320*B*x^3*a^2*b^(7/2)*(x*(
b*x+a))^(1/2)+5952*A*x^2*a^2*b^(7/2)*(x*(b*x+a))^(1/2)+80*B*x^2*a^3*b^(5/2)*(x*(b*x+a))^(1/2)+240*A*(x*(b*x+a)
)^(1/2)*b^(5/2)*x*a^3-100*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a^4+180*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)
/b^(1/2))*a^5*b-360*A*(x*(b*x+a))^(1/2)*b^(3/2)*a^4-75*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))
*a^6+150*B*(x*(b*x+a))^(1/2)*b^(1/2)*a^5)/(x*(b*x+a))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.76362, size = 849, normalized size = 3.77 \begin{align*} \left [-\frac{15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \,{\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \,{\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \,{\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \,{\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{15360 \, b^{4}}, \frac{15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \,{\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \,{\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \,{\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \,{\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{7680 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="fricas")

[Out]

[-1/15360*(15*(5*B*a^6 - 12*A*a^5*b)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(1280*B*b^6*
x^5 + 75*B*a^5*b - 180*A*a^4*b^2 + 128*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 + 8*(
5*B*a^3*b^3 + 372*A*a^2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^4, 1/7680*(15*(
5*B*a^6 - 12*A*a^5*b)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (1280*B*b^6*x^5 + 75*B*a^5*b - 180
*A*a^4*b^2 + 128*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 + 8*(5*B*a^3*b^3 + 372*A*a^
2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^4]

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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(x**(3/2)*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

Timed out

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Giac [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(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="giac")

[Out]

Timed out