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

Optimal. Leaf size=192 \[ -\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-7 a B)}{192 b^3}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-7 a B)}{128 b^4}-\frac{a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{9/2}}+\frac{a x^{5/2} \sqrt{a+b x} (10 A b-7 a B)}{240 b^2}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-7 a B)}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b} \]

[Out]

(a^3*(10*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^4) - (a^2*(10*A*b - 7*a*B)*x^(3/2)*Sqrt[a + b*x])/(192*b^3
) + (a*(10*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(240*b^2) + ((10*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b) + (
B*x^(7/2)*(a + b*x)^(3/2))/(5*b) - (a^4*(10*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(9/2
))

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Rubi [A]  time = 0.0906541, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-7 a B)}{192 b^3}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-7 a B)}{128 b^4}-\frac{a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{9/2}}+\frac{a x^{5/2} \sqrt{a+b x} (10 A b-7 a B)}{240 b^2}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-7 a B)}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(10*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^4) - (a^2*(10*A*b - 7*a*B)*x^(3/2)*Sqrt[a + b*x])/(192*b^3
) + (a*(10*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(240*b^2) + ((10*A*b - 7*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b) + (
B*x^(7/2)*(a + b*x)^(3/2))/(5*b) - (a^4*(10*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(9/2
))

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^{5/2} \sqrt{a+b x} (A+B x) \, dx &=\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac{\left (5 A b-\frac{7 a B}{2}\right ) \int x^{5/2} \sqrt{a+b x} \, dx}{5 b}\\ &=\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac{(a (10 A b-7 a B)) \int \frac{x^{5/2}}{\sqrt{a+b x}} \, dx}{80 b}\\ &=\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac{\left (a^2 (10 A b-7 a B)\right ) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{96 b^2}\\ &=-\frac{a^2 (10 A b-7 a B) x^{3/2} \sqrt{a+b x}}{192 b^3}+\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac{\left (a^3 (10 A b-7 a B)\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{128 b^3}\\ &=\frac{a^3 (10 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{128 b^4}-\frac{a^2 (10 A b-7 a B) x^{3/2} \sqrt{a+b x}}{192 b^3}+\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac{\left (a^4 (10 A b-7 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{256 b^4}\\ &=\frac{a^3 (10 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{128 b^4}-\frac{a^2 (10 A b-7 a B) x^{3/2} \sqrt{a+b x}}{192 b^3}+\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac{\left (a^4 (10 A b-7 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{128 b^4}\\ &=\frac{a^3 (10 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{128 b^4}-\frac{a^2 (10 A b-7 a B) x^{3/2} \sqrt{a+b x}}{192 b^3}+\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac{\left (a^4 (10 A b-7 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^4}\\ &=\frac{a^3 (10 A b-7 a B) \sqrt{x} \sqrt{a+b x}}{128 b^4}-\frac{a^2 (10 A b-7 a B) x^{3/2} \sqrt{a+b x}}{192 b^3}+\frac{a (10 A b-7 a B) x^{5/2} \sqrt{a+b x}}{240 b^2}+\frac{(10 A b-7 a B) x^{7/2} \sqrt{a+b x}}{40 b}+\frac{B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac{a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.258524, size = 146, normalized size = 0.76 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (-4 a^2 b^2 x (25 A+14 B x)+10 a^3 b (15 A+7 B x)-105 a^4 B+16 a b^3 x^2 (5 A+3 B x)+96 b^4 x^3 (5 A+4 B x)\right )+\frac{15 a^{7/2} (7 a B-10 A b) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{1920 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*(-105*a^4*B + 16*a*b^3*x^2*(5*A + 3*B*x) + 96*b^4*x^3*(5*A + 4*B*x) + 10*a^3*b
*(15*A + 7*B*x) - 4*a^2*b^2*x*(25*A + 14*B*x)) + (15*a^(7/2)*(-10*A*b + 7*a*B)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[
a]])/Sqrt[1 + (b*x)/a]))/(1920*b^(9/2))

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Maple [A]  time = 0.013, size = 260, normalized size = 1.4 \begin{align*} -{\frac{1}{3840}\sqrt{x}\sqrt{bx+a} \left ( -768\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-960\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-96\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-160\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+112\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+200\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x{a}^{2}-140\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{3}+150\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{4}b-300\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{3}-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{5}+210\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{4} \right ){b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(b*x+a)^(1/2),x)

[Out]

-1/3840*x^(1/2)*(b*x+a)^(1/2)/b^(9/2)*(-768*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)-960*A*x^3*b^(9/2)*(x*(b*x+a))^(1/2
)-96*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)-160*A*x^2*a*b^(7/2)*(x*(b*x+a))^(1/2)+112*B*x^2*a^2*b^(5/2)*(x*(b*x+a))
^(1/2)+200*A*(x*(b*x+a))^(1/2)*b^(5/2)*x*a^2-140*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a^3+150*A*ln(1/2*(2*(x*(b*x+a))
^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^4*b-300*A*(x*(b*x+a))^(1/2)*b^(3/2)*a^3-105*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b
^(1/2)+2*b*x+a)/b^(1/2))*a^5+210*B*(x*(b*x+a))^(1/2)*b^(1/2)*a^4)/(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^(5/2)*(B*x+A)*(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.6906, size = 728, normalized size = 3.79 \begin{align*} \left [-\frac{15 \,{\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt{b} \log \left (2 \, b x - 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \,{\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \,{\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \,{\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3840 \, b^{5}}, -\frac{15 \,{\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (384 \, B b^{5} x^{4} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \,{\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \,{\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \,{\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{1920 \, b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/3840*(15*(7*B*a^5 - 10*A*a^4*b)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(384*B*b^5*x^
4 - 105*B*a^4*b + 150*A*a^3*b^2 + 48*(B*a*b^4 + 10*A*b^5)*x^3 - 8*(7*B*a^2*b^3 - 10*A*a*b^4)*x^2 + 10*(7*B*a^3
*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^5, -1/1920*(15*(7*B*a^5 - 10*A*a^4*b)*sqrt(-b)*arctan(sqrt(b*
x + a)*sqrt(-b)/(b*sqrt(x))) - (384*B*b^5*x^4 - 105*B*a^4*b + 150*A*a^3*b^2 + 48*(B*a*b^4 + 10*A*b^5)*x^3 - 8*
(7*B*a^2*b^3 - 10*A*a*b^4)*x^2 + 10*(7*B*a^3*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^5]

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Sympy [C]  time = 75.1845, size = 2382, normalized size = 12.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

2*A*a**2*Piecewise((a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(b*x/a)) - 3*sqrt(a)*(a + b*x)**(3/2)/(8*sqrt(b)*sqr
t(b*x/a)) - a**2*acosh(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b)) + (a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(b*x/a)),
Abs(a + b*x)/Abs(a) > 1), (-I*a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(-b*x/a)) + 3*I*sqrt(a)*(a + b*x)**(3/2)/(
8*sqrt(b)*sqrt(-b*x/a)) + I*a**2*asin(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b)) - I*(a + b*x)**(5/2)/(4*sqrt(a)*sqrt(
b)*sqrt(-b*x/a)), True))/b**3 - 4*A*a*Piecewise((a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a
+ b*x)**(3/2)/(48*sqrt(b)*sqrt(b*x/a)) - 5*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(b*x/a)) - a**3*acosh(sqr
t(a + b*x)/sqrt(a))/(16*sqrt(b)) + (a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(a + b*x)/Abs(a) > 1),
(-I*a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(3/2)/(48*sqrt(b)*sqrt(-b*x/a))
+ 5*I*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(-b*x/a)) + I*a**3*asin(sqrt(a + b*x)/sqrt(a))/(16*sqrt(b)) - I
*(a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3 + 2*A*Piecewise((5*a**(7/2)*sqrt(a + b*x)/(128
*sqrt(b)*sqrt(b*x/a)) - 5*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(5/2)/(192
*sqrt(b)*sqrt(b*x/a)) - 7*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sqrt(b*x/a)) - 5*a**4*acosh(sqrt(a + b*x)/sqrt(
a))/(128*sqrt(b)) + (a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(a + b*x)/Abs(a) > 1), (-5*I*a**(7/2)
*sqrt(a + b*x)/(128*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(-b*x/a)) + I*a**(3
/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(-b*x/a)) + 7*I*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I
*a**4*asin(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) - I*(a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/
b**3 - 2*B*a**3*Piecewise((a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(b*x/a)) - 3*sqrt(a)*(a + b*x)**(3/2)/(8*sqrt
(b)*sqrt(b*x/a)) - a**2*acosh(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b)) + (a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(b*
x/a)), Abs(a + b*x)/Abs(a) > 1), (-I*a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(-b*x/a)) + 3*I*sqrt(a)*(a + b*x)**
(3/2)/(8*sqrt(b)*sqrt(-b*x/a)) + I*a**2*asin(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b)) - I*(a + b*x)**(5/2)/(4*sqrt(a
)*sqrt(b)*sqrt(-b*x/a)), True))/b**4 + 6*B*a**2*Piecewise((a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(b*x/a)) - a
**(3/2)*(a + b*x)**(3/2)/(48*sqrt(b)*sqrt(b*x/a)) - 5*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(b*x/a)) - a**3
*acosh(sqrt(a + b*x)/sqrt(a))/(16*sqrt(b)) + (a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(a + b*x)/Ab
s(a) > 1), (-I*a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(3/2)/(48*sqrt(b)*sqrt
(-b*x/a)) + 5*I*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(-b*x/a)) + I*a**3*asin(sqrt(a + b*x)/sqrt(a))/(16*sq
rt(b)) - I*(a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**4 - 6*B*a*Piecewise((5*a**(7/2)*sqrt(a
+ b*x)/(128*sqrt(b)*sqrt(b*x/a)) - 5*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)
**(5/2)/(192*sqrt(b)*sqrt(b*x/a)) - 7*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sqrt(b*x/a)) - 5*a**4*acosh(sqrt(a
+ b*x)/sqrt(a))/(128*sqrt(b)) + (a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(a + b*x)/Abs(a) > 1), (-
5*I*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(-b*x/a
)) + I*a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(-b*x/a)) + 7*I*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sqrt(-b
*x/a)) + 5*I*a**4*asin(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) - I*(a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(-b*x/
a)), True))/b**4 + 2*B*Piecewise((7*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(b*x/a)) - 7*a**(7/2)*(a + b*x)**(
3/2)/(768*sqrt(b)*sqrt(b*x/a)) - 7*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**
(7/2)/(480*sqrt(b)*sqrt(b*x/a)) - 9*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(b*x/a)) - 7*a**5*acosh(sqrt(a +
b*x)/sqrt(a))/(256*sqrt(b)) + (a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(a + b*x)/Abs(a) > 1), (-
7*I*a**(9/2)*sqrt(a + b*x)/(256*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**(7/2)*(a + b*x)**(3/2)/(768*sqrt(b)*sqrt(-b*x/a
)) + 7*I*a**(5/2)*(a + b*x)**(5/2)/(1920*sqrt(b)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(7/2)/(480*sqrt(b)*sqrt
(-b*x/a)) + 9*I*sqrt(a)*(a + b*x)**(9/2)/(80*sqrt(b)*sqrt(-b*x/a)) + 7*I*a**5*asin(sqrt(a + b*x)/sqrt(a))/(256
*sqrt(b)) - I*(a + b*x)**(11/2)/(10*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**4

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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^(5/2)*(B*x+A)*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out