\(\int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx\) [657]

Optimal result

Integrand size = 20, antiderivative size = 315 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \]

[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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {(5 a d+7 b c) (b c-a d)^5 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}-\frac {(a+b x)^{7/2} \sqrt {c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]

[In]

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

[Out]

-1/512*((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(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])/(
960*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 52

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 65

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 81

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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*} \text {integral}& = \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 ) \text {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 ) \text {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*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.93 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (49 c+10 d x)+10 a^3 b^2 d^3 \left (15 c^2+16 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-91 c^3+58 c^2 d x+564 c d^2 x^2+360 d^3 x^3\right )+a b^4 d \left (415 c^4-272 c^3 d x+216 c^2 d^2 x^2+4448 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^4}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(49*c + 10*d*x) + 10*a^3*b^2*d^3*(15*c^2 + 16*c*d*x + 4
*d^2*x^2) + 6*a^2*b^3*d^2*(-91*c^3 + 58*c^2*d*x + 564*c*d^2*x^2 + 360*d^3*x^3) + a*b^4*d*(415*c^4 - 272*c^3*d*
x + 216*c^2*d^2*x^2 + 4448*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(-105*c^5 + 70*c^4*d*x - 56*c^3*d^2*x^2 + 48*c^2*d^
3*x^3 + 1664*c*d^4*x^4 + 1280*d^5*x^5)))/(7680*b^3*d^4) + ((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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(265)=530\).

Time = 1.59 (sec) , antiderivative size = 1037, normalized size of antiderivative = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1037\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.84 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\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 ] \]

[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)]

Sympy [F]

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

[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)

Maxima [F(-2)]

Exception generated. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]

[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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2375 vs. \(2 (265) = 530\).

Time = 0.61 (sec) , antiderivative size = 2375, normalized size of antiderivative = 7.54 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \]

[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

Mupad [F(-1)]

Timed out. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]

[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)