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

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

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Rubi [A]  time = 0.25, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \begin {gather*} -\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} \end {gather*}

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 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 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 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])

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]

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*}

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Mathematica [A]  time = 3.11, size = 377, normalized size = 1.08 \begin {gather*} \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} \end {gather*}

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)

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IntegrateAlgebraic [A]  time = 0.77, size = 397, normalized size = 1.14 \begin {gather*} \frac {5 (b c-a d)^6 (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac {\sqrt {c+d x} (b c-a d)^6 \left (\frac {105 b^7 c (c+d x)^6}{(a+b x)^6}+\frac {105 a b^6 d (c+d x)^6}{(a+b x)^6}-\frac {700 b^6 c d (c+d x)^5}{(a+b x)^5}-\frac {700 a b^5 d^2 (c+d x)^5}{(a+b x)^5}+\frac {1981 b^5 c d^2 (c+d x)^4}{(a+b x)^4}+\frac {1981 a b^4 d^3 (c+d x)^4}{(a+b x)^4}-\frac {3072 b^4 c d^3 (c+d x)^3}{(a+b x)^3}+\frac {3072 a b^3 d^4 (c+d x)^3}{(a+b x)^3}-\frac {1981 b^3 c d^4 (c+d x)^2}{(a+b x)^2}-\frac {1981 a b^2 d^5 (c+d x)^2}{(a+b x)^2}+\frac {700 b^2 c d^5 (c+d x)}{a+b x}+\frac {700 a b d^6 (c+d x)}{a+b x}-105 a d^7-105 b c d^6\right )}{21504 b^4 d^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21504*((b*c - a*d)^6*Sqrt[c + d*x]*(-105*b*c*d^6 - 105*a*d^7 + (700*b^2*c*d^5*(c + d*x))/(a + b*x) + (700*a
*b*d^6*(c + d*x))/(a + b*x) - (1981*b^3*c*d^4*(c + d*x)^2)/(a + b*x)^2 - (1981*a*b^2*d^5*(c + d*x)^2)/(a + b*x
)^2 - (3072*b^4*c*d^3*(c + d*x)^3)/(a + b*x)^3 + (3072*a*b^3*d^4*(c + d*x)^3)/(a + b*x)^3 + (1981*b^5*c*d^2*(c
+ d*x)^4)/(a + b*x)^4 + (1981*a*b^4*d^3*(c + d*x)^4)/(a + b*x)^4 - (700*b^6*c*d*(c + d*x)^5)/(a + b*x)^5 - (7
00*a*b^5*d^2*(c + d*x)^5)/(a + b*x)^5 + (105*b^7*c*(c + d*x)^6)/(a + b*x)^6 + (105*a*b^6*d*(c + d*x)^6)/(a + b
*x)^6))/(b^4*d^4*Sqrt[a + b*x]*(-d + (b*(c + d*x))/(a + b*x))^7) + (5*(b*c - a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[
b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(1024*b^(9/2)*d^(9/2))

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fricas [A]  time = 1.11, size = 1102, normalized size = 3.17 \begin {gather*} \left [\frac {105 \, {\left (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}\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 (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 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (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}\right )} x^{3} - 8 \, {\left (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}\right )} x^{2} + 2 \, {\left (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}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{86016 \, b^{5} d^{5}}, -\frac {105 \, {\left (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}\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 (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 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (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}\right )} x^{3} - 8 \, {\left (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}\right )} x^{2} + 2 \, {\left (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}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{43008 \, b^{5} d^{5}}\right ] \end {gather*}

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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giac [B]  time = 6.66, size = 4150, normalized size = 11.93

result too large to display

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*(1680*(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))*a*c^2*abs(b) + 13440*(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))*a^2*c^2*abs(b)/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^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7
*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a
^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21
*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6
*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*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^3*d^4))*b*c^2*abs(b) + 336*(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^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26
*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 44
7*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*
b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 +
10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*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^3*d^4))*a*c*d*abs(b) + 8960*(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) + s
qrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a^3*c*d*abs(b)/b^2 + 3360*(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
))*a^2*c*d*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^
5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^3
4*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x +
a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)
/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8
+ 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^
2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(abs(-sqrt(b*d)*sq
rt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*b*c*d*abs(b) + 42*(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^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10
)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*
d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7
+ 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^
5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b
^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*
b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*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^4*d^5))*a*d^2*abs(b) + 560*(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^1
1*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))*s
qrt(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))*a^3*d^2*abs(b)/b^2 + 168*(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^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b
^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*
c^2*d^6 + 37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5
+ 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a
*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*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^3*d^4))*a^2*d^2*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^6 + (b^42*c*d^11 - 85*a*b^41*d^12)/(b^47*d^12
)) - (11*b^43*c^2*d^10 + 62*a*b^42*c*d^11 - 2593*a^2*b^41*d^12)/(b^47*d^12)) + 3*(33*b^44*c^3*d^9 + 153*a*b^43
*c^2*d^10 + 435*a^2*b^42*c*d^11 - 11821*a^3*b^41*d^12)/(b^47*d^12))*(b*x + a) - 7*(33*b^45*c^4*d^8 + 120*a*b^4
4*c^3*d^9 + 282*a^2*b^43*c^2*d^10 + 544*a^3*b^42*c*d^11 - 10579*a^4*b^41*d^12)/(b^47*d^12))*(b*x + a) + 35*(33
*b^46*c^5*d^7 + 87*a*b^45*c^4*d^8 + 162*a^2*b^44*c^3*d^9 + 262*a^3*b^43*c^2*d^10 + 397*a^4*b^42*c*d^11 - 5549*
a^5*b^41*d^12)/(b^47*d^12))*(b*x + a) - 105*(33*b^47*c^6*d^6 + 54*a*b^46*c^5*d^7 + 75*a^2*b^45*c^4*d^8 + 100*a
^3*b^44*c^3*d^9 + 135*a^4*b^43*c^2*d^10 + 198*a^5*b^42*c*d^11 - 1619*a^6*b^41*d^12)/(b^47*d^12))*sqrt(b*x + a)
- 105*(33*b^7*c^7 + 21*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 63*a^5*b^
2*c^2*d^5 + 231*a^6*b*c*d^6 - 429*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^5*d^6))*b*d^2*abs(b) + 26880*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5
*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2
*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*a^3*c^2*abs(b)/b^3)/b

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maple [B]  time = 0.02, size = 1580, normalized size = 4.54

result too large to display

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)*(980*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^5*b*c*d^5+105*b^7*c^7*l
n(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+105*a^7*d^7*ln(1/2*(2*b*d*x
+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-525*a^6*b*c*d^6*ln(1/2*(2*b*d*x+a*d+b*c+2
*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+9472*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^
2*b^4*d^6*x^4+9472*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^2*d^4*x^4+96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)*a^3*b^3*d^6*x^3+96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^3*d^3*x^3-112*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^4*b^2*d^6*x^2-112*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^4*d^2*x
^2+980*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^5*d-210*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)
*a^6*d^6-210*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^6-1582*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1
/2)*a^4*b^2*c^2*d^4+600*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3-1582*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c^4*d^2+140*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^5*b*d^6*x+140*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x+14848*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*d^6*x
^5+14848*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*c*d^5*x^5+945*a^5*b^2*c^2*d^5*ln(1/2*(2*b*d*x+a*d+b*c
+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-525*a^4*b^3*c^3*d^4*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-525*a^3*b^4*c^4*d^3*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+945*a^2*b^5*c^5*d^2*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))-525*a*b^6*c^6*d*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))+6144*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^6*d^6*x^6-644*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^4*d^2*x-644*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^4*b^2*c*d^5*
x+1016*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2*d^4*x+1016*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d
)^(1/2)*a^2*b^4*c^3*d^3*x+37376*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c*d^5*x^4+25504*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^4*c*d^5*x^3+25504*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^2*
d^4*x^3+512*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^3*b^3*c*d^5*x^2+19680*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*(b*d)^(1/2)*a^2*b^4*c^2*d^4*x^2+512*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^5*c^3*d^3*x^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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