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

Optimal. Leaf size=306 \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{96 b^4 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{64 b^5 d}-\frac{5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-\frac{63 a^2 d}{b}+14 a c+\frac{b c^2}{d}\right )}{24 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 \sqrt{a+b x} (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d} \]

[Out]

(-5*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^5*d) - (5*(b^2*c^2 + 14
*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^4*d) - ((14*a*c + (b*c^2)/d - (63*a^2*d)/b)*Sqrt[a
+ b*x]*(c + d*x)^(5/2))/(24*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*x]) + (Sqr
t[a + b*x]*(c + d*x)^(7/2))/(4*b^2*d) - (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(11/2)*d^(3/2))

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Rubi [A]  time = 0.333521, antiderivative size = 306, 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, 80, 50, 63, 217, 206} \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{96 b^4 d}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{64 b^5 d}-\frac{5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-\frac{63 a^2 d}{b}+14 a c+\frac{b c^2}{d}\right )}{24 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 \sqrt{a+b x} (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^5*d) - (5*(b^2*c^2 + 14
*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^4*d) - ((14*a*c + (b*c^2)/d - (63*a^2*d)/b)*Sqrt[a
+ b*x]*(c + d*x)^(5/2))/(24*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*x]) + (Sqr
t[a + b*x]*(c + d*x)^(7/2))/(4*b^2*d) - (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(11/2)*d^(3/2))

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 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 \frac{x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx &=-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{2 \int \frac{(c+d x)^{5/2} \left (-\frac{1}{2} a (b c-7 a d)+\frac{1}{2} b (b c-a d) x\right )}{\sqrt{a+b x}} \, dx}{b^2 (b c-a d)}\\ &=-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \int \frac{(c+d x)^{5/2}}{\sqrt{a+b x}} \, dx}{8 b^2 d (b c-a d)}\\ &=-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^3 d}\\ &=-\frac{5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{64 b^4 d}\\ &=-\frac{5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^5 d}-\frac{5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^5 d}\\ &=-\frac{5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^5 d}-\frac{5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 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 )}{64 b^6 d}\\ &=-\frac{5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^5 d}-\frac{5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 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 )}{64 b^6 d}\\ &=-\frac{5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^5 d}-\frac{5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac{\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac{5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.890378, size = 245, normalized size = 0.8 \[ \frac{\sqrt{c+d x} \left (\frac{\sqrt{d} \left (a^2 b^2 d \left (-839 c^2+637 c d x+126 d^2 x^2\right )+105 a^3 b d^2 (17 c-3 d x)-945 a^4 d^3+a b^3 \left (-337 c^2 d x+15 c^3-244 c d^2 x^2-72 d^3 x^3\right )+b^4 x \left (118 c^2 d x+15 c^3+136 c d^2 x^2+48 d^3 x^3\right )\right )}{\sqrt{a+b x}}-\frac{15 (b c-a d)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{192 b^5 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*((Sqrt[d]*(-945*a^4*d^3 + 105*a^3*b*d^2*(17*c - 3*d*x) + a^2*b^2*d*(-839*c^2 + 637*c*d*x + 126*
d^2*x^2) + a*b^3*(15*c^3 - 337*c^2*d*x - 244*c*d^2*x^2 - 72*d^3*x^3) + b^4*x*(15*c^3 + 118*c^2*d*x + 136*c*d^2
*x^2 + 48*d^3*x^3)))/Sqrt[a + b*x] - (15*(b*c - a*d)^(3/2)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcSinh[(Sqrt[d
]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(192*b^5*d^(3/2))

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Maple [B]  time = 0.026, size = 961, 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*(d*x+c)^(5/2)/(b*x+a)^(3/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(96*x^4*b^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-144*x^3*a*b^3*d^3*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+272*x^3*b^4*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+945*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^4*b*d^4-2100*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*b^2*c*d^3+1350*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^3*c^2*d^2-180*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^4*c^3*d-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*b^5*c^4+252*x^2*a
^2*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-488*x^2*a*b^3*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+236*x^2
*b^4*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+945*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*d^4-2100*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*c*d^3+1350*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^2*c^2*d^2-180*l
n(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^3*c^3*d-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*b^4*c^4-630*x*a^3*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+1274*x*a^2*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-674*x*a*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+30*x*b^4*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1890*a^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3
570*a^3*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1678*a^2*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*
a*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^5/d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.34411, size = 1752, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 12*
a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 - 63*a^4*b*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*(48*b^5*d^4*x^4 + 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785*a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5
*c*d^3 - 9*a*b^4*d^4)*x^3 + 2*(59*b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d^4)*x^2 + (15*b^5*c^3*d - 337*a*
b^4*c^2*d^2 + 637*a^2*b^3*c*d^3 - 315*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2), 1/
384*(15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 12*a*b^
4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 - 63*a^4*b*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*(48*b^5*d^4*x^4
+ 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785*a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5*c*d^3 - 9*a*b^4*d^4)*x
^3 + 2*(59*b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d^4)*x^2 + (15*b^5*c^3*d - 337*a*b^4*c^2*d^2 + 637*a^2*b
^3*c*d^3 - 315*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2)]

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

[Out]

Timed out

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Giac [A]  time = 2.3504, size = 625, normalized size = 2.04 \begin{align*} \frac{1}{192} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{7}} + \frac{17 \, b^{28} c d^{7}{\left | b \right |} - 33 \, a b^{27} d^{8}{\left | b \right |}}{b^{34} d^{6}}\right )} + \frac{59 \, b^{29} c^{2} d^{6}{\left | b \right |} - 326 \, a b^{28} c d^{7}{\left | b \right |} + 315 \, a^{2} b^{27} d^{8}{\left | b \right |}}{b^{34} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{30} c^{3} d^{5}{\left | b \right |} - 191 \, a b^{29} c^{2} d^{6}{\left | b \right |} + 511 \, a^{2} b^{28} c d^{7}{\left | b \right |} - 325 \, a^{3} b^{27} d^{8}{\left | b \right |}\right )}}{b^{34} d^{6}}\right )} \sqrt{b x + a} - \frac{4 \,{\left (\sqrt{b d} a^{2} b^{3} c^{3}{\left | b \right |} - 3 \, \sqrt{b d} a^{3} b^{2} c^{2} d{\left | b \right |} + 3 \, \sqrt{b d} a^{4} b c d^{2}{\left | b \right |} - \sqrt{b d} a^{5} d^{3}{\left | b \right |}\right )}}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} b^{6}} + \frac{5 \,{\left (\sqrt{b d} b^{4} c^{4}{\left | b \right |} + 12 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} - 90 \, \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} + 140 \, \sqrt{b d} a^{3} b c d^{3}{\left | b \right |} - 63 \, \sqrt{b d} a^{4} d^{4}{\left | b \right |}\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 )}^{2}\right )}{128 \, b^{7} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*d^2*abs(b)/b^7 + (17*b^28*c*d
^7*abs(b) - 33*a*b^27*d^8*abs(b))/(b^34*d^6)) + (59*b^29*c^2*d^6*abs(b) - 326*a*b^28*c*d^7*abs(b) + 315*a^2*b^
27*d^8*abs(b))/(b^34*d^6)) + 3*(5*b^30*c^3*d^5*abs(b) - 191*a*b^29*c^2*d^6*abs(b) + 511*a^2*b^28*c*d^7*abs(b)
- 325*a^3*b^27*d^8*abs(b))/(b^34*d^6))*sqrt(b*x + a) - 4*(sqrt(b*d)*a^2*b^3*c^3*abs(b) - 3*sqrt(b*d)*a^3*b^2*c
^2*d*abs(b) + 3*sqrt(b*d)*a^4*b*c*d^2*abs(b) - sqrt(b*d)*a^5*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*b^6) + 5/128*(sqrt(b*d)*b^4*c^4*abs(b) + 12*sqrt(b*d)*a*b^3*c^
3*d*abs(b) - 90*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) + 140*sqrt(b*d)*a^3*b*c*d^3*abs(b) - 63*sqrt(b*d)*a^4*d^4*abs
(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^7*d^2)