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

Optimal. Leaf size=319 \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{11/2}}+\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

[Out]

(2*c^2*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(5*b*c - 3*a*d)*(a + b*x)^(7/2))/(3*d^2*(b*
c - a*d)^2*Sqrt[c + d*x]) + (5*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (5*(
21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^4*(b*c - a*d)) + ((21*b^2*c^2 - 14*a*b
*c*d + a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^3*(b*c - a*d)^2) - (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d
+ a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.351792, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {89, 78, 50, 63, 217, 206} \[ \frac{(a+b x)^{5/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{11/2}}+\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^2*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(5*b*c - 3*a*d)*(a + b*x)^(7/2))/(3*d^2*(b*
c - a*d)^2*Sqrt[c + d*x]) + (5*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (5*(
21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^4*(b*c - a*d)) + ((21*b^2*c^2 - 14*a*b
*c*d + a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^3*(b*c - a*d)^2) - (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d
+ a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(11/2))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

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

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 \frac{x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{2 \int \frac{(a+b x)^{5/2} \left (\frac{1}{2} c (7 b c-3 a d)-\frac{3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{d^2 (b c-a d)^2}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}-\frac{\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{6 d^3 (b c-a d)}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}-\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^4 (b c-a d)}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}+\frac{\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d^4}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^4 (b c-a d)}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}-\frac{\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^5}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^4 (b c-a d)}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}-\frac{\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\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 )}{8 b d^5}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^4 (b c-a d)}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}-\frac{\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\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 )}{8 b d^5}\\ &=\frac{2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac{4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt{c+d x}}+\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 d^5}-\frac{5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^4 (b c-a d)}+\frac{\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt{c+d x}}{3 d^3 (b c-a d)^2}-\frac{5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{11/2}}\\ \end{align*}

Mathematica [A]  time = 1.46044, size = 282, normalized size = 0.88 \[ \frac{\frac{15 (c+d x)^2 (b c-a d)^3 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \left (\frac{16 d^3 (a+b x)^3}{15 (b c-a d)^3}-\frac{4 d^2 (a+b x)^2}{3 (b c-a d)^2}+\frac{2 d (a+b x)}{b c-a d}-\frac{2 \sqrt{d} \sqrt{a+b x} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{4 d^4 (a d-b c)}-8 c^2 (a+b x)^4+\frac{16 c (a+b x)^4 (c+d x) (5 b c-3 a d)}{b c-a d}}{12 d^2 \sqrt{a+b x} (c+d x)^{3/2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c^2*(a + b*x)^4 + (16*c*(5*b*c - 3*a*d)*(a + b*x)^4*(c + d*x))/(b*c - a*d) + (15*(b*c - a*d)^3*(21*b^2*c^2
- 14*a*b*c*d + a^2*d^2)*(c + d*x)^2*((2*d*(a + b*x))/(b*c - a*d) - (4*d^2*(a + b*x)^2)/(3*(b*c - a*d)^2) + (1
6*d^3*(a + b*x)^3)/(15*(b*c - a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*
d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(4*d^4*(-(b*c) + a*d)))/(12*d^2*(-(b*c) + a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))

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Maple [B]  time = 0.027, size = 1002, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(b*x+a)^(1/2)*(16*x^4*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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))*x^2*a^3*d^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))*x^2*a^2*b*c*d^4+525*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))*x^2*a*b^2*c^2*d^3-315*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))*x^2*
b^3*c^3*d^2+52*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b*d^4-36*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*b^2*
c*d^3+30*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))*x*a^3*c*d^4-450*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))*x*a^2*b*c^2*d^3+1050*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))*x*a*b^2*c^3*d^2-630*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))*x*b^3*c^4*d+66*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^2*d^4-192*(b*
d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a*b*c*d^3+126*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*b^2*c^2*d^2+15*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*c^2*d^3-225*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^2*b*c^3*d^2+525*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^2*c^4*d-315*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^3*c^5+324*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*c*d^3-1148*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*x*a*b*c^2*d^2+840*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*b^2*c^3*d+226*(b*d)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)*a^2*c^2*d^2-840*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^3*d+630*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b
^2*c^4)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(d*x+c)^(3/2)/d^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.3874, size = 1769, normalized size = 5.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3
+ 15*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(21*b^3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*d^4)*x)*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*(8*b^3*d^5*x^4 + 315*b^3*c^4*d - 420*a*b^2*c^3*d^2 + 113*a^2*b*c^2*d^3
- 2*(9*b^3*c*d^4 - 13*a*b^2*d^5)*x^3 + 3*(21*b^3*c^2*d^3 - 32*a*b^2*c*d^4 + 11*a^2*b*d^5)*x^2 + 2*(210*b^3*c^3
*d^2 - 287*a*b^2*c^2*d^3 + 81*a^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^8*x^2 + 2*b*c*d^7*x + b*c^2*d^
6), 1/48*(15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^
3 + 15*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(21*b^3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*d^4)*x)*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*(8*b^3*d^5*x^4 + 315*b^3*c^4*d - 420*a*b^2*c^3*d^2 + 113*a^2*b*c^2*d^3 - 2*(9*b^3*c*d^4
- 13*a*b^2*d^5)*x^3 + 3*(21*b^3*c^2*d^3 - 32*a*b^2*c*d^4 + 11*a^2*b*d^5)*x^2 + 2*(210*b^3*c^3*d^2 - 287*a*b^2*
c^2*d^3 + 81*a^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^8*x^2 + 2*b*c*d^7*x + b*c^2*d^6)]

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

[Out]

Timed out

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Giac [A]  time = 1.42566, size = 694, normalized size = 2.18 \begin{align*} \frac{{\left ({\left ({\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b^{6} c d^{8} - a b^{5} d^{9}\right )}{\left (b x + a\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}} - \frac{3 \,{\left (3 \, b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} - a^{2} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )} + \frac{3 \,{\left (21 \, b^{8} c^{3} d^{6} - 35 \, a b^{7} c^{2} d^{7} + 15 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{20 \,{\left (21 \, b^{9} c^{4} d^{5} - 56 \, a b^{8} c^{3} d^{6} + 50 \, a^{2} b^{7} c^{2} d^{7} - 16 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (21 \, b^{10} c^{5} d^{4} - 77 \, a b^{9} c^{4} d^{5} + 106 \, a^{2} b^{8} c^{3} d^{6} - 66 \, a^{3} b^{7} c^{2} d^{7} + 17 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{4} c d^{9}{\left | b \right |} - a b^{3} d^{10}{\left | b \right |}}\right )} \sqrt{b x + a}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (21 \, b^{4} c^{3} - 35 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt{b d} d^{5}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(((2*(b*x + a)*(4*(b^6*c*d^8 - a*b^5*d^9)*(b*x + a)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)) - 3*(3*b^7*c^2
*d^7 - 2*a*b^6*c*d^8 - a^2*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b))) + 3*(21*b^8*c^3*d^6 - 35*a*b^7*c^2
*d^7 + 15*a^2*b^6*c*d^8 - a^3*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*(b*x + a) + 20*(21*b^9*c^4*d^5
- 56*a*b^8*c^3*d^6 + 50*a^2*b^7*c^2*d^7 - 16*a^3*b^6*c*d^8 + a^4*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b
)))*(b*x + a) + 15*(21*b^10*c^5*d^4 - 77*a*b^9*c^4*d^5 + 106*a^2*b^8*c^3*d^6 - 66*a^3*b^7*c^2*d^7 + 17*a^4*b^6
*c*d^8 - a^5*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3
/2) + 5/8*(21*b^4*c^3 - 35*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 - a^3*b*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)*d^5*abs(b))