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

Optimal. Leaf size=348 \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^5}{1024 b^4 d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^4}{1536 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{384 b^4 d^2}+\frac{5 (a d+b c) (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (a d+b c) (b c-a d)}{24 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (a d+b c)}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d} \]

[Out]

(-5*(b*c - a*d)^5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^4) + (5*(b*c - a*d)^4*(b*c + a*d)*(a +
b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^3) - ((b*c - a*d)^3*(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^4*
d^2) - ((b*c - a*d)^2*(b*c + a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(b*c + a*d)*(a + b*
x)^(7/2)*(c + d*x)^(3/2))/(24*b^3*d) - ((b*c + a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(12*b^2*d) + ((a + b*x)^(
7/2)*(c + d*x)^(7/2))/(7*b*d) + (5*(b*c - a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(1024*b^(9/2)*d^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.248056, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^5}{1024 b^4 d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^4}{1536 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{384 b^4 d^2}+\frac{5 (a d+b c) (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (a d+b c) (b c-a d)}{24 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (a d+b c)}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)^5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^4) + (5*(b*c - a*d)^4*(b*c + a*d)*(a +
b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^3) - ((b*c - a*d)^3*(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^4*
d^2) - ((b*c - a*d)^2*(b*c + a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(b*c + a*d)*(a + b*
x)^(7/2)*(c + d*x)^(3/2))/(24*b^3*d) - ((b*c + a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(12*b^2*d) + ((a + b*x)^(
7/2)*(c + d*x)^(7/2))/(7*b*d) + (5*(b*c - a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(1024*b^(9/2)*d^(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 (a+b x)^{5/2} (c+d x)^{5/2} \, dx &=\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac{(b c+a d) \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx}{2 b d}\\ &=-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac{\left (5 \left (c^2-\frac{a^2 d^2}{b^2}\right )\right ) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{24 d}\\ &=-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac{\left ((b c-a d)^2 (b c+a d)\right ) \int (a+b x)^{5/2} \sqrt{c+d x} \, dx}{16 b^3 d}\\ &=-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac{\left ((b c-a d)^3 (b c+a d)\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{128 b^4 d}\\ &=-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac{\left (5 (b c-a d)^4 (b c+a d)\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{768 b^4 d^2}\\ &=\frac{5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^3}-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac{\left (5 (b c-a d)^5 (b c+a d)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{1024 b^4 d^3}\\ &=-\frac{5 (b c-a d)^5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^4}+\frac{5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^3}-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac{\left (5 (b c-a d)^6 (b c+a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2048 b^4 d^4}\\ &=-\frac{5 (b c-a d)^5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^4}+\frac{5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^3}-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac{\left (5 (b c-a d)^6 (b c+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 )}{1024 b^5 d^4}\\ &=-\frac{5 (b c-a d)^5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^4}+\frac{5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^3}-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac{\left (5 (b c-a d)^6 (b c+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 )}{1024 b^5 d^4}\\ &=-\frac{5 (b c-a d)^5 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{1024 b^4 d^4}+\frac{5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt{c+d x}}{1536 b^4 d^3}-\frac{(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt{c+d x}}{384 b^4 d^2}-\frac{(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt{c+d x}}{64 b^4 d}-\frac{(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac{(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac{5 (b c-a d)^6 (b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 3.1447, size = 377, normalized size = 1.08 \[ \frac{(a+b x)^{7/2} (c+d x)^{7/2} \left (7-\frac{49 \sqrt{b c-a d} (a d+b c) \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \left (16 d^{7/2} (a+b x)^4 (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 a^2 d^2-2 a b d (7 c+4 d x)+b^2 \left (27 c^2+40 c d x+16 d^2 x^2\right )\right )-10 d^{3/2} (a+b x)^2 (b c-a d)^{11/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+8 d^{5/2} (a+b x)^3 (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+15 \sqrt{d} (a+b x) (b c-a d)^{13/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-15 \sqrt{a+b x} (b c-a d)^7 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{3072 b^5 d^{7/2} (a+b x)^4 (c+d x)^5}\right )}{49 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(7/2)*(c + d*x)^(7/2)*(7 - (49*Sqrt[b*c - a*d]*(b*c + a*d)*((b*(c + d*x))/(b*c - a*d))^(3/2)*(15*Sq
rt[d]*(b*c - a*d)^(13/2)*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)] - 10*d^(3/2)*(b*c - a*d)^(11/2)*(a + b*x)^2
*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 8*d^(5/2)*(b*c - a*d)^(9/2)*(a + b*x)^3*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 1
6*d^(7/2)*(b*c - a*d)^(3/2)*(a + b*x)^4*Sqrt[(b*(c + d*x))/(b*c - a*d)]*(3*a^2*d^2 - 2*a*b*d*(7*c + 4*d*x) + b
^2*(27*c^2 + 40*c*d*x + 16*d^2*x^2)) - 15*(b*c - a*d)^7*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c
- a*d]]))/(3072*b^5*d^(7/2)*(a + b*x)^4*(c + d*x)^5)))/(49*b*d)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 1580, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43008*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*b*d^6-210*(b*d)^(1/
2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^6*d^6-210*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^6*c^6+6144*x^6*b^
6*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^6*c^5*d+
980*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*b*c*d^5-1582*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a
^4*b^2*c^2*d^4+600*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^3*c^3*d^3-1582*(b*d)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*a^2*b^4*c^4*d^2+980*a*b^5*c^5*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+14848*x^5*a*b^5*
d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+14848*x^5*b^6*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)+9472*x^4*a^2*b^4*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+9472*x^4*b^6*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)+96*x^3*a^3*b^3*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*b^6*c^3*d^3*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-112*x^2*a^4*b^2*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-112*x^2*
b^6*c^4*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-525*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^6*b*c*d^6+945*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*b^2*c^2*d^5-525*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^3*c^3*d^4-525*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^4*c^4*d^3+945*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^5*c^5*d^2-525*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^6*c^6*d+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^7*d^7+10
5*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^7*c^7-644*(b*d)^(1/2)*
(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b^2*c*d^5+1016*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b^3*c^2
*d^4+1016*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b^4*c^3*d^3-644*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*x*a*b^5*c^4*d^2+37376*x^4*a*b^5*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+25504*x^3*a^2*b^4*
c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+25504*x^3*a*b^5*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d
)^(1/2)+512*x^2*a^3*b^3*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+19680*x^2*a^2*b^4*c^2*d^4*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+512*x^2*a*b^5*c^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/b^4/d^4/(
b*d*x^2+a*d*x+b*c*x+a*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*(b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.69204, size = 2411, normalized size = 6.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/86016*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2
*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*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*(3072*b^7*d^7*x^6 - 105*b^7*c
^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d^6
- 105*a^6*b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 146*a*b^6*c*d^6 + 37*a^2*b^5*d^7)*
x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^5 + 797*a^2*b^5*c*d^6 + 3*a^3*b^4*d^7)*x^3 - 8*(7*b^7*c^4*d^3 - 32*a
*b^6*c^3*d^4 - 1230*a^2*b^5*c^2*d^5 - 32*a^3*b^4*c*d^6 + 7*a^4*b^3*d^7)*x^2 + 2*(35*b^7*c^5*d^2 - 161*a*b^6*c^
4*d^3 + 254*a^2*b^5*c^3*d^4 + 254*a^3*b^4*c^2*d^5 - 161*a^4*b^3*c*d^6 + 35*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(b^5*d^5), -1/43008*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^
3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*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*(3072*b^7*d^7*x^6 - 105*b^7
*c^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d
^6 - 105*a^6*b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 146*a*b^6*c*d^6 + 37*a^2*b^5*d^7
)*x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^5 + 797*a^2*b^5*c*d^6 + 3*a^3*b^4*d^7)*x^3 - 8*(7*b^7*c^4*d^3 - 32
*a*b^6*c^3*d^4 - 1230*a^2*b^5*c^2*d^5 - 32*a^3*b^4*c*d^6 + 7*a^4*b^3*d^7)*x^2 + 2*(35*b^7*c^5*d^2 - 161*a*b^6*
c^4*d^3 + 254*a^2*b^5*c^3*d^4 + 254*a^3*b^4*c^2*d^5 - 161*a^4*b^3*c*d^6 + 35*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqr
t(d*x + c))/(b^5*d^5)]

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

[Out]

Timed out

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Giac [B]  time = 2.05544, size = 4644, normalized size = 13.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*(56*(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))*c^2*abs(b) + 1120*(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(-sqr
t(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*a*c^2*abs(b)/b + 28*(sqrt(b^2*
c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*c*d^9 - 49*a*b^20*d^10)/(
b^24*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) + (21*b^23*c^3*d^7 + 49*a*
b^22*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21*b^24*c^4*d^6 + 28*a*b^23
*c^3*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*(b*x + a) + 15*(21*b^25*c
^5*d^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^9 - 21*a^5*b^20*d^10)/(b^
24*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 5*a^4*b^2*
c^2*d^4 - 14*a^5*b*c*d^5 + 21*a^6*d^6)*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^3*d^5))*c*d*abs(b) + 1120*(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))*a^2*c*d*abs(b)/b^2 + 224*(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 + 1
6*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))*a*c*d*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(8*(b*x + a)*(10*(b*x + a)*(12*(b*x + a)/b
^5 + (b^31*c*d^11 - 71*a*b^30*d^12)/(b^35*d^12)) - (11*b^32*c^2*d^10 + 48*a*b^31*c*d^11 - 1739*a^2*b^30*d^12)/
(b^35*d^12)) + 3*(33*b^33*c^3*d^9 + 111*a*b^32*c^2*d^10 + 239*a^2*b^31*c*d^11 - 5983*a^3*b^30*d^12)/(b^35*d^12
))*(b*x + a) - 7*(33*b^34*c^4*d^8 + 78*a*b^33*c^3*d^9 + 128*a^2*b^32*c^2*d^10 + 178*a^3*b^31*c*d^11 - 3617*a^4
*b^30*d^12)/(b^35*d^12))*(b*x + a) + 35*(33*b^35*c^5*d^7 + 45*a*b^34*c^4*d^8 + 50*a^2*b^33*c^3*d^9 + 50*a^3*b^
32*c^2*d^10 + 45*a^4*b^31*c*d^11 - 991*a^5*b^30*d^12)/(b^35*d^12))*(b*x + a) - 105*(33*b^36*c^6*d^6 + 12*a*b^3
5*c^5*d^7 + 5*a^2*b^34*c^4*d^8 - 5*a^4*b^32*c^2*d^10 - 12*a^5*b^31*c*d^11 - 33*a^6*b^30*d^12)/(b^35*d^12))*sqr
t(b*x + a) - 105*(33*b^7*c^7 - 21*a*b^6*c^6*d - 7*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 - 7*
a^5*b^2*c^2*d^5 - 21*a^6*b*c*d^6 + 33*a^7*d^7)*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^4*d^6))*d^2*abs(b) + 56*(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(a
bs(-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))*a^2*d^2*abs(b)/b^2 +
28*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^4 + (b^21*c*d^9 - 49*a
*b^20*d^10)/(b^24*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) + (21*b^23*c^
3*d^7 + 49*a*b^22*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21*b^24*c^4*d^
6 + 28*a*b^23*c^3*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*(b*x + a) +
15*(21*b^25*c^5*d^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^9 - 21*a^5*b
^20*d^10)/(b^24*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3
- 5*a^4*b^2*c^2*d^4 - 14*a^5*b*c*d^5 + 21*a^6*d^6)*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^3*d^5))*a*d^2*abs(b)/b + 56*(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))*a^2*c^2*abs(b)/b^3)/b