3.657 \(\int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=315 \[ \frac {(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}-\frac {(a+b x)^{7/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]

[Out]

-1/60*(5*a*d+7*b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2/d+1/6*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b/d+1/512*(-a*d+b*c)^5*(
5*a*d+7*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(9/2)+1/768*(-a*d+b*c)^3*(5*a*d+7*
b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^3-1/960*(-a*d+b*c)^2*(5*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d^2-
1/160*(-a*d+b*c)*(5*a*d+7*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^3/d-1/512*(-a*d+b*c)^4*(5*a*d+7*b*c)*(b*x+a)^(1/2
)*(d*x+c)^(1/2)/b^3/d^4

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Rubi [A]  time = 0.20, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}+\frac {(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac {(a+b x)^{7/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

-((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^4) + ((b*c - a*d)^3*(7*b*c + 5*a*d)*(a
+ b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^3) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(96
0*b^3*d^2) - ((b*c - a*d)*(7*b*c + 5*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d) - ((7*b*c + 5*a*d)*(a + b
*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(5/2))/(6*b*d) + ((b*c - a*d)^5*(7*b*c + 5*
a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*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)^{3/2} \, dx &=\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(7 b c+5 a d) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{12 b d}\\ &=-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {((b c-a d) (7 b c+5 a d)) \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{40 b^2 d}\\ &=-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{320 b^3 d}\\ &=-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {\left ((b c-a d)^3 (7 b c+5 a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{384 b^3 d^2}\\ &=\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {\left ((b c-a d)^4 (7 b c+5 a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^3 d^3}\\ &=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {\left ((b c-a d)^5 (7 b c+5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^3 d^4}\\ &=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {\left ((b c-a d)^5 (7 b c+5 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 )}{512 b^4 d^4}\\ &=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {\left ((b c-a d)^5 (7 b c+5 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 )}{512 b^4 d^4}\\ &=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {(b c-a d)^5 (7 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 2.15, size = 348, normalized size = 1.10 \[ \frac {(a+b x)^{7/2} (c+d x)^{5/2} \left (7-\frac {7 \sqrt {b c-a d} (5 a d+7 b c) \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \left (-10 d^{3/2} (a+b x)^2 (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}}+8 d^{5/2} (a+b x)^3 (b c-a d)^{7/2} \sqrt {\frac {b (c+d x)}{b c-a d}}+16 d^{7/2} (a+b x)^4 (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} (-3 a d+11 b c+8 b d x)+15 \sqrt {d} (a+b x) (b c-a d)^{11/2} \sqrt {\frac {b (c+d x)}{b c-a d}}-15 \sqrt {a+b x} (b c-a d)^6 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{1280 b^4 d^{7/2} (a+b x)^4 (c+d x)^4}\right )}{42 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

((a + b*x)^(7/2)*(c + d*x)^(5/2)*(7 - (7*Sqrt[b*c - a*d]*(7*b*c + 5*a*d)*((b*(c + d*x))/(b*c - a*d))^(3/2)*(15
*Sqrt[d]*(b*c - a*d)^(11/2)*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)] - 10*d^(3/2)*(b*c - a*d)^(9/2)*(a + b*x)
^2*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 8*d^(5/2)*(b*c - a*d)^(7/2)*(a + b*x)^3*Sqrt[(b*(c + d*x))/(b*c - a*d)] +
16*d^(7/2)*(b*c - a*d)^(3/2)*(a + b*x)^4*Sqrt[(b*(c + d*x))/(b*c - a*d)]*(11*b*c - 3*a*d + 8*b*d*x) - 15*(b*c
- a*d)^6*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]))/(1280*b^4*d^(7/2)*(a + b*x)^4*(c +
d*x)^4)))/(42*b*d)

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fricas [A]  time = 1.13, size = 894, normalized size = 2.84 \[ \left [-\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} - 105 \, b^{6} c^{5} d + 415 \, a b^{5} c^{4} d^{2} - 546 \, a^{2} b^{4} c^{3} d^{3} + 150 \, a^{3} b^{3} c^{2} d^{4} - 245 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (13 \, b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 278 \, a b^{5} c d^{5} + 135 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 27 \, a b^{5} c^{2} d^{4} - 423 \, a^{2} b^{4} c d^{5} - 5 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 136 \, a b^{5} c^{3} d^{3} + 174 \, a^{2} b^{4} c^{2} d^{4} + 80 \, a^{3} b^{3} c d^{5} - 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{4} d^{5}}, -\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} - 105 \, b^{6} c^{5} d + 415 \, a b^{5} c^{4} d^{2} - 546 \, a^{2} b^{4} c^{3} d^{3} + 150 \, a^{3} b^{3} c^{2} d^{4} - 245 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (13 \, b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 278 \, a b^{5} c d^{5} + 135 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 27 \, a b^{5} c^{2} d^{4} - 423 \, a^{2} b^{4} c d^{5} - 5 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 136 \, a b^{5} c^{3} d^{3} + 174 \, a^{2} b^{4} c^{2} d^{4} + 80 \, a^{3} b^{3} c d^{5} - 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{4} d^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/30720*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*
a^5*b*c*d^5 - 5*a^6*d^6)*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*(1280*b^6*d^6*x^5 - 105*b^6*c^5*d + 415*
a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 150*a^3*b^3*c^2*d^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d
^5 + 25*a*b^5*d^6)*x^4 + 16*(3*b^6*c^2*d^4 + 278*a*b^5*c*d^5 + 135*a^2*b^4*d^6)*x^3 - 8*(7*b^6*c^3*d^3 - 27*a*
b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^3*d^6)*x^2 + 2*(35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2
*d^4 + 80*a^3*b^3*c*d^5 - 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^5), -1/15360*(15*(7*b^6*c^6 -
30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*s
qrt(-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*(1280*b^6*d^6*x^5 - 105*b^6*c^5*d + 415*a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 150*a^3
*b^3*c^2*d^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d^5 + 25*a*b^5*d^6)*x^4 + 16*(3*b^6*c^2*d^4 +
278*a*b^5*c*d^5 + 135*a^2*b^4*d^6)*x^3 - 8*(7*b^6*c^3*d^3 - 27*a*b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^3*d
^6)*x^2 + 2*(35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2*d^4 + 80*a^3*b^3*c*d^5 - 25*a^4*b^2*d^6)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^5)]

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giac [B]  time = 4.48, size = 2375, normalized size = 7.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="giac")

[Out]

1/7680*(120*(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*abs(b) + 960*(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*abs(b)/b + 4*(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*abs(b) + 12*(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*d*abs(b) + 320*(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^3*d*abs(b)/b^2 + 120*(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*d*abs(b)/b +
(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 - 7
93*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))*b*d*abs(b) + 1920*(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*abs(b)/b^3)/b

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maple [B]  time = 0.02, size = 1240, normalized size = 3.94 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-2560 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} d^{5} x^{5}-6400 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} d^{5} x^{4}-3328 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c \,d^{4} x^{4}+75 a^{6} d^{6} \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 )-270 a^{5} b c \,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 )+225 a^{4} b^{2} c^{2} 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 )+300 a^{3} b^{3} c^{3} 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 )-675 a^{2} b^{4} c^{4} 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 )-4320 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} d^{5} x^{3}+450 a \,b^{5} c^{5} 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 )-8896 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c \,d^{4} x^{3}-105 b^{6} c^{6} \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}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{2} d^{3} x^{3}-80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} d^{5} x^{2}-6768 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c \,d^{4} x^{2}-432 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{2} d^{3} x^{2}+112 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{3} d^{2} x^{2}+100 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,d^{5} x -320 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c \,d^{4} x -696 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{2} d^{3} x +544 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{3} d^{2} x -140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{4} d x -150 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5}+490 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4}-300 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3}+1092 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2}-830 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d +210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{5}\right )}{15360 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{3} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x)

[Out]

-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-6400*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*d^5*x^4-3328*(b*
d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c*d^4*x^4-4320*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^
3*d^5*x^3-96*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^2*d^3*x^3-80*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*a^3*b^2*d^5*x^2+112*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^3*d^2*x^2+100*(b*d)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*d^5*x-140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^4*d*x+490*(b*d)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c*d^4-300*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^2*
d^3+1092*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*c^3*d^2-830*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*a*b^4*c^4*d-270*a^5*b*c*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))+225*a^4*b^2*c^2*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
))+300*a^3*b^3*c^3*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))-675
*a^2*b^4*c^4*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))+450*a*b^5
*c^5*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))-2560*(b*d)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*d^5*x^5-150*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*d^5+210*(b*d)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^5+75*a^6*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))-105*b^6*c^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))-8896*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*c*d^4*x^3-6768*(b*d)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*a^2*b^3*c*d^4*x^2-432*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*c^2*d^3*x^2-320*(b*
d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c*d^4*x-696*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b
^3*c^2*d^3*x+544*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*c^3*d^2*x)/b^3/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 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),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 \[ \int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x)

[Out]

int(x*(a + b*x)^(5/2)*(c + d*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)

[Out]

Integral(x*(a + b*x)**(5/2)*(c + d*x)**(3/2), x)

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