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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.195259, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {80, 50, 63, 217, 206} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}+\frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int x (a+b x)^{5/2} (c+d x)^{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*}

Mathematica [A]  time = 1.99253, size = 348, normalized size = 1.1 \[ \frac{(a+b x)^{7/2} (c+d x)^{5/2} \left (7-\frac{7 \sqrt{b c-a d} (5 a d+7 b c) \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \left (-10 d^{3/2} (a+b x)^2 (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+8 d^{5/2} (a+b x)^3 (b c-a d)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}}+16 d^{7/2} (a+b x)^4 (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} (-3 a d+11 b c+8 b d x)+15 \sqrt{d} (a+b x) (b c-a d)^{11/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-15 \sqrt{a+b x} (b c-a d)^6 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{1280 b^4 d^{7/2} (a+b x)^4 (c+d x)^4}\right )}{42 b d} \]

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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Maple [B]  time = 0.016, size = 1240, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^3*b^5*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-8896*x^3
*a*b^4*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-320*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3
*b^2*c*d^4-696*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b^3*c^2*d^3+544*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*x*a*b^4*c^3*d^2-6768*x^2*a^2*b^3*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-432*x^2*a*b^
4*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-2560*x^5*b^5*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(
1/2)-80*x^2*a^3*b^2*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+112*x^2*b^5*c^3*d^2*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)-6400*x^4*a*b^4*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-3328*x^4*b^5*c*d^4*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-4320*x^3*a^2*b^3*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+100*(b*
d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*d^5-140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^5*c^4
*d+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+75*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d
)^(1/2))*a^6*d^6-105*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^6*c
^6-150*(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^5-270*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c*d^5+225*l
n(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^2*d^4+300*ln(1/2*
(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^3*d^3-675*ln(1/2*(2*b*d
*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^4*d^2+450*ln(1/2*(2*b*d*x+2*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^5*d)/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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.93833, size = 1989, normalized size = 6.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(d*x+c)^(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.39443, size = 2682, normalized size = 8.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*(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^3 + (b^13*c*d^7 - 3
1*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263*a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*
d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d^8)/(b^15*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a
*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 - 5*a*b^4*c^4*d -
2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^
2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^4))*c*abs(b) + 80*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x
+ a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 -
59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqr
t(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*log(abs(-sqrt(b*d)*
sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^3))*a*c*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^4 + (b^21*c*d^9 - 49*a*b^20*d^10)/(b^24*d^10))
- 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) + (21*b^23*c^3*d^7 + 49*a*b^22*c^2*d^8
+ 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21*b^24*c^4*d^6 + 28*a*b^23*c^3*d^7 + 3
0*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*(b*x + a) + 15*(21*b^25*c^5*d^5 + 7*a
*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^9 - 21*a^5*b^20*d^10)/(b^24*d^10))*sq
rt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 14
*a^5*b*c*d^5 + 21*a^6*d^6)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)
*b^3*d^5))*d*abs(b) + 40*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^
7*c*d^5 - 17*a*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^
3*d^3 + a*b^8*c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d
- 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*
d - a*b*d)))/(sqrt(b*d)*b*d^3))*a^2*d*abs(b)/b^2 + 8*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(
b*x + a)*(8*(b*x + a)/b^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263
*a^2*b^12*d^8)/(b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d^8)/(b^15
*d^8))*(b*x + a) - 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt
(b*x + a) - 15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)
*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^4))*a*d*abs(b)/b +
4*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/(b^6*d^2) + (b*c*d^3 - 7*a*d^4)
/(b^6*d^6)) - 3*(b^2*c^2*d^2 - a^2*d^4)/(b^6*d^6)) - 3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*log(abs
(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^5*d^4))*a^2*c*abs(b)/b^3)/b