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

Optimal. Leaf size=268 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}+\frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]

[Out]

-((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^4*d^2) - ((b*c - a*d)^2*(3*b*c + 7*a*d)*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*b^
3*d) - ((3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(40*b^2*d) + ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*b*d
) + ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(5
/2))

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Rubi [A]  time = 0.157163, antiderivative size = 268, 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} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}+\frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^4*d^2) - ((b*c - a*d)^2*(3*b*c + 7*a*d)*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*b^
3*d) - ((3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(40*b^2*d) + ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*b*d
) + ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(5
/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 \sqrt{a+b x} (c+d x)^{5/2} \, dx &=\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac{(3 b c+7 a d) \int \sqrt{a+b x} (c+d x)^{5/2} \, dx}{10 b d}\\ &=-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac{((b c-a d) (3 b c+7 a d)) \int \sqrt{a+b x} (c+d x)^{3/2} \, dx}{16 b^2 d}\\ &=-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac{\left ((b c-a d)^2 (3 b c+7 a d)\right ) \int \sqrt{a+b x} \sqrt{c+d x} \, dx}{32 b^3 d}\\ &=-\frac{(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac{\left ((b c-a d)^3 (3 b c+7 a d)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b^4 d}\\ &=-\frac{(b c-a d)^3 (3 b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^2}-\frac{(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^4 (3 b c+7 a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^4 d^2}\\ &=-\frac{(b c-a d)^3 (3 b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^2}-\frac{(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^4 (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 )}{128 b^5 d^2}\\ &=-\frac{(b c-a d)^3 (3 b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^2}-\frac{(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac{\left ((b c-a d)^4 (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 )}{128 b^5 d^2}\\ &=-\frac{(b c-a d)^3 (3 b c+7 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^2}-\frac{(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac{(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac{(b c-a d)^4 (3 b c+7 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.34095, size = 291, normalized size = 1.09 \[ \frac{(a+b x)^{3/2} (c+d x)^{7/2} \left (3-\frac{(7 a d+3 b c) \sqrt{\frac{b (c+d x)}{b c-a d}} \left (2 b^5 d^2 (a+b x)^2 (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (5 c+2 d x)+b^2 \left (59 c^2+68 c d x+24 d^2 x^2\right )\right )+15 b^5 d (a+b x) (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-15 b^5 \sqrt{d} \sqrt{a+b x} (b c-a d)^5 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{128 b^9 d^2 (a+b x)^2 (c+d x)^4 \sqrt{b c-a d}}\right )}{15 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(3/2)*(c + d*x)^(7/2)*(3 - ((3*b*c + 7*a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*(15*b^5*d*(b*c - a*d)^(
9/2)*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 2*b^5*d^2*(b*c - a*d)^(3/2)*(a + b*x)^2*Sqrt[(b*(c + d*x))/(b
*c - a*d)]*(15*a^2*d^2 - 10*a*b*d*(5*c + 2*d*x) + b^2*(59*c^2 + 68*c*d*x + 24*d^2*x^2)) - 15*b^5*Sqrt[d]*(b*c
- a*d)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]))/(128*b^9*d^2*Sqrt[b*c - a*d]*(a + b*
x)^2*(c + d*x)^4)))/(15*b*d)

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Maple [B]  time = 0.013, size = 942, normalized size = 3.5 \begin{align*}{\frac{1}{3840\,{d}^{2}{b}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+2016\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+352\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+1488\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-375\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}+450\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-444\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}+436\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+60\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+680\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}-692\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+120\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-90\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\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]

1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(768*x^4*b^4*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*a*b^3*d
^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+2016*x^3*b^4*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-
112*x^2*a^2*b^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+352*x^2*a*b^3*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*(b*d)^(1/2)+1488*x^2*b^4*c^2*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*d^5-375*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*c*d^4+450*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c^2*d^3-150*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c^3*d^2-75*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^4*d+45*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*b^5*c^5+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b*d^4-444*(b*d)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x*a^2*b^2*c*d^3+436*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^3*c^2*d^2+60*(b*d)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^4*c^3*d-210*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4+680*(b
*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3-692*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*
c^2*d^2+120*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d-90*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*b^4*c^4)/d^2/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^4/(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 = 2.98172, size = 1571, normalized size = 5.86 \begin{align*} \left [\frac{15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \,{\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, b^{5} d^{3}}, -\frac{15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \,{\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \,{\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, b^{5} d^{3}}\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]

[1/7680*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)
*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*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 60*a*b^4*c^3*d^2 - 346*a^2*b^3
*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(21*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 22*a*b^
4*c*d^4 - 7*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*a^2*b^3*c*d^4 + 35*a^3*b^2*d^5)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^3), -1/3840*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b
^2*c^2*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)*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*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 60*a*b^4*
c^3*d^2 - 346*a^2*b^3*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(21*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*
b^5*c^2*d^3 + 22*a*b^4*c*d^4 - 7*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*a^2*b^3*c*d^4
+ 35*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^3)]

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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 = 1.41445, size = 1153, normalized size = 4.3 \begin{align*} \text{result too large to display} \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]

1/1920*(20*(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^2 + (b^7*c*d^5 - 17*a
*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*
c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + 5*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*d^3))*c*d*abs(b)/b^2 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x +
a)/b^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263*a^2*b^12*d^8)/(b^
15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d^8)/(b^15*d^8))*(b*x + a) -
15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt(b*x + a) - 15*(7*
b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*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^4))*d^2*abs(b)/b^2 + (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^6*d^2) + (b*c*d^3 - 7*a*d^4)/(b^6*d^6)) - 3*(b
^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 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^5*d^4))*c^2*abs(b)/b^3)/b