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

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

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Rubi [A]  time = 0.16, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {\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} \end {gather*}

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 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 \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*}

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Mathematica [A]  time = 1.63, size = 291, normalized size = 1.09 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.48, size = 306, normalized size = 1.14 \begin {gather*} \frac {(b c-a d)^4 (7 a d+3 b c) \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 {\sqrt {a+b x} (b c-a d)^4 \left (-\frac {210 b^4 c d (a+b x)}{c+d x}+105 a b^4 d+\frac {790 a b^3 d^2 (a+b x)}{c+d x}-\frac {384 b^3 c d^2 (a+b x)^2}{(c+d x)^2}-\frac {896 a b^2 d^3 (a+b x)^2}{(c+d x)^2}+\frac {210 b^2 c d^3 (a+b x)^3}{(c+d x)^3}-\frac {105 a d^5 (a+b x)^4}{(c+d x)^4}+\frac {490 a b d^4 (a+b x)^3}{(c+d x)^3}-\frac {45 b c d^4 (a+b x)^4}{(c+d x)^4}+45 b^5 c\right )}{1920 b^4 d^2 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1920*((b*c - a*d)^4*Sqrt[a + b*x]*(45*b^5*c + 105*a*b^4*d - (45*b*c*d^4*(a + b*x)^4)/(c + d*x)^4 - (105*a*d
^5*(a + b*x)^4)/(c + d*x)^4 + (210*b^2*c*d^3*(a + b*x)^3)/(c + d*x)^3 + (490*a*b*d^4*(a + b*x)^3)/(c + d*x)^3
- (384*b^3*c*d^2*(a + b*x)^2)/(c + d*x)^2 - (896*a*b^2*d^3*(a + b*x)^2)/(c + d*x)^2 - (210*b^4*c*d*(a + b*x))/
(c + d*x) + (790*a*b^3*d^2*(a + b*x))/(c + d*x)))/(b^4*d^2*Sqrt[c + d*x]*(b - (d*(a + b*x))/(c + d*x))^5) + ((
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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fricas [A]  time = 1.61, size = 704, normalized size = 2.63 \begin {gather*} \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 {gather*}

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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giac [B]  time = 2.83, size = 1522, normalized size = 5.68

result too large to display

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*(80*(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^2*abs(b)/b + 160*(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*c*d*abs(b)/b^2 + 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^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))*c*d*abs(b)/b + 10*(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*d^2*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^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))*d^2*abs(b)/b + 480*(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*c^2*abs(b)/b^3)/b

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maple [B]  time = 0.02, size = 942, normalized size = 3.51 \begin {gather*} \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+105 a^{5} d^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-375 a^{4} b c \,d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+450 a^{3} b^{2} c^{2} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-150 a^{2} b^{3} c^{3} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-75 a \,b^{4} c^{4} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}+45 b^{5} c^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}-112 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}+352 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,d^{4} x -444 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c \,d^{3} x +436 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{2} d^{2} x +60 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{3} d x -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4}+680 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3}-692 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2}+120 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d -90 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{2}} \end {gather*}

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*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^4*d^4*x^4+96*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a*b^3*d^4*x^3+2016*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^4*c*d^3*x^3-
112*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*b^2*d^4*x^2+352*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)*a*b^3*c*d^3*x^2+1488*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*b^4*c^2*d^2*x^2+105*a^5*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))-375*a^4*b*c*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))+450*a^3*b^2*c^2*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))-150*a^2*b^3*c^3*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))-75*a*b^4*c^4*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))+45*b^5*c^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))+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*d^4*x-444*(b*d)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*a^2*b^2*c*d^3*x+436*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^2*d^2*x+60*(b*d)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^4*c^3*d*x-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 = 0.00, 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*(d*x+c)^(5/2)*(b*x+a)^(1/2),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\,\sqrt {a+b\,x}\,{\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)^(1/2)*(c + d*x)^(5/2),x)

[Out]

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

[Out]

Timed out

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