3.715 \(\int \frac{x (c+d x)^{5/2}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=217 \[ -\frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 b^2 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (7 a d+b c)}{96 b^3 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d} \]

[Out]

(-5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^4*d) - (5*(b*c - a*d)*(b*c + 7*a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(96*b^3*d) - ((b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b^2*d) + (Sqrt[a + b*x]
*(c + d*x)^(7/2))/(4*b*d) - (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d
*x])])/(64*b^(9/2)*d^(3/2))

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Rubi [A]  time = 0.121698, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 b^2 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (7 a d+b c)}{96 b^3 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^4*d) - (5*(b*c - a*d)*(b*c + 7*a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(96*b^3*d) - ((b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b^2*d) + (Sqrt[a + b*x]
*(c + d*x)^(7/2))/(4*b*d) - (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d
*x])])/(64*b^(9/2)*d^(3/2))

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 \frac{x (c+d x)^{5/2}}{\sqrt{a+b x}} \, dx &=\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{(b c+7 a d) \int \frac{(c+d x)^{5/2}}{\sqrt{a+b x}} \, dx}{8 b d}\\ &=-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{(5 (b c-a d) (b c+7 a d)) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^2 d}\\ &=-\frac{5 (b c-a d) (b c+7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{\left (5 (b c-a d)^2 (b c+7 a d)\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{64 b^3 d}\\ &=-\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d}-\frac{5 (b c-a d) (b c+7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{\left (5 (b c-a d)^3 (b c+7 a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^4 d}\\ &=-\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d}-\frac{5 (b c-a d) (b c+7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{\left (5 (b c-a d)^3 (b c+7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{64 b^5 d}\\ &=-\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d}-\frac{5 (b c-a d) (b c+7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{\left (5 (b c-a d)^3 (b c+7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 b^5 d}\\ &=-\frac{5 (b c-a d)^2 (b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{64 b^4 d}-\frac{5 (b c-a d) (b c+7 a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac{(b c+7 a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d}-\frac{5 (b c-a d)^3 (b c+7 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.552033, size = 194, normalized size = 0.89 \[ \frac{\sqrt{c+d x} \left (\sqrt{d} \sqrt{a+b x} \left (5 a^2 b d^2 (53 c+14 d x)-105 a^3 d^3-a b^2 d \left (191 c^2+172 c d x+56 d^2 x^2\right )+b^3 \left (118 c^2 d x+15 c^3+136 c d^2 x^2+48 d^3 x^3\right )\right )-\frac{15 (b c-a d)^{5/2} (7 a d+b c) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{192 b^4 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*(Sqrt[d]*Sqrt[a + b*x]*(-105*a^3*d^3 + 5*a^2*b*d^2*(53*c + 14*d*x) - a*b^2*d*(191*c^2 + 172*c*d
*x + 56*d^2*x^2) + b^3*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3)) - (15*(b*c - a*d)^(5/2)*(b*c + 7*a
*d)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(192*b^4*d^(3/2))

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Maple [B]  time = 0.019, size = 574, normalized size = 2.7 \begin{align*}{\frac{1}{384\,{b}^{4}d}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+272\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-300\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+270\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}{c}^{3}d-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}{c}^{4}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}-344\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}+236\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-210\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}+530\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}-382\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d+30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}{c}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^(5/2)/(b*x+a)^(1/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(96*x^3*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-112*x^2*a*b^2*d^3*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+272*x^2*b^3*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^4-300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c*d^3+270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)
/(b*d)^(1/2))*a^2*b^2*c^2*d^2-60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a
*b^3*c^3*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^4+140*(b*d)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b*d^3-344*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a*b^2*c*d^2+236*(b*d)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)*x*b^3*c^2*d-210*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3+530*(b*d)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)*a^2*b*c*d^2-382*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d+30*(b*d)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)*b^3*c^3)/b^4/d/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(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*(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.3407, size = 1231, normalized size = 5.67 \begin{align*} \left [-\frac{15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \,{\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{5} d^{2}}, \frac{15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \,{\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \,{\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b^{5} d^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*sqrt(b*d)*log(8*b^2*d^
2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2
*c*d + a*b*d^2)*x) - 4*(48*b^4*d^4*x^3 + 15*b^4*c^3*d - 191*a*b^3*c^2*d^2 + 265*a^2*b^2*c*d^3 - 105*a^3*b*d^4
+ 8*(17*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(59*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x + a)*s
qrt(d*x + c))/(b^5*d^2), 1/384*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)
*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (
b^2*c*d + a*b*d^2)*x)) + 2*(48*b^4*d^4*x^3 + 15*b^4*c^3*d - 191*a*b^3*c^2*d^2 + 265*a^2*b^2*c*d^3 - 105*a^3*b*
d^4 + 8*(17*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(59*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(b^5*d^2)]

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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*(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.46898, size = 875, normalized size = 4.03 \begin{align*} \frac{\frac{16 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{2}}\right )} c d{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac{5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}}\right )} d^{2}{\left | b \right |}}{b^{2}} + \frac{4 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - 5 \, a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} c^{2}{\left | b \right |}}{b^{3}}}{192 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/192*(16*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a
*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*
d)*b*d^2))*c*d*abs(b)/b^2 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 +
(b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) +
3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*
b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) +
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*d^2*abs(b)/b^2 + 4*(sqrt(b^2*c + (b*x + a)*b*d - a*
b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^4*d^2) + (b*c*d - 5*a*d^2)/(b^4*d^4)) + (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*l
og(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^3))*c^2*abs(b)/b^3)/b