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

Optimal. Leaf size=246 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{8 b^4 d}-\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-\frac{35 a^2 d}{b}+10 a c+\frac{b c^2}{d}\right )}{12 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 \sqrt{a+b x} (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d} \]

[Out]

-((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4*d) - ((10*a*c + (b*c^2)/d - (35*a^2*
d)/b)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(5/2))/(b^2*(b*c - a*d)*Sqrt[a +
b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b^2*d) - ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(
Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(9/2)*d^(3/2))

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Rubi [A]  time = 0.26803, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{8 b^4 d}-\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-\frac{35 a^2 d}{b}+10 a c+\frac{b c^2}{d}\right )}{12 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 \sqrt{a+b x} (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^4*d) - ((10*a*c + (b*c^2)/d - (35*a^2*
d)/b)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(12*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(5/2))/(b^2*(b*c - a*d)*Sqrt[a +
b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b^2*d) - ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(
Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(9/2)*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)^{3/2}}{(a+b x)^{3/2}} \, dx &=-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{2 \int \frac{(c+d x)^{3/2} \left (-\frac{1}{2} a (b c-5 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)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{6 b^2 d (b c-a d)}\\ &=-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{8 b^3 d}\\ &=-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b^4 d}\\ &=-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 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^5 d}\\ &=-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 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^5 d}\\ &=-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{8 b^4 d}-\frac{\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt{a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac{(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.50918, size = 194, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (\frac{\sqrt{d} \left (5 a^2 b d (7 d x-20 c)+105 a^3 d^2+a b^2 \left (3 c^2-38 c d x-14 d^2 x^2\right )+b^3 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{\sqrt{a+b x}}-\frac{3 \sqrt{b c-a d} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{24 b^4 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*((Sqrt[d]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x) + a*b^2*(3*c^2 - 38*c*d*x - 14*d^2*x^2) + b^
3*x*(3*c^2 + 14*c*d*x + 8*d^2*x^2)))/Sqrt[a + b*x] - (3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Ar
cSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(24*b^4*d^(3/2))

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Maple [B]  time = 0.022, size = 692, normalized size = 2.8 \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)^(3/2)/(b*x+a)^(3/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(-16*x^3*b^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*x*a^3*b*d^3-135*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^2*c*d^2+27*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^3*c^2*d+3*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^4*c^
3+28*x^2*a*b^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-28*x^2*b^3*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*a^4*d^3-135*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*c*d^2+27*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^2*c^2*d+3*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^3*c^3-70*x*a^2*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+76*x*a*b^2*c*d*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)-6*x*b^3*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-210*a^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+200*a^2*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a*b^2*c^2*((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^4/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)^(3/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.60269, size = 1314, normalized size = 5.34 \begin{align*} \left [\frac{3 \,{\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\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 (8 \, b^{4} d^{3} x^{3} + 3 \, a b^{3} c^{2} d - 100 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 14 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} +{\left (3 \, b^{4} c^{2} d - 38 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}}, \frac{3 \,{\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\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 (8 \, b^{4} d^{3} x^{3} + 3 \, a b^{3} c^{2} d - 100 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 14 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} +{\left (3 \, b^{4} c^{2} d - 38 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*
d^2 + 35*a^3*b*d^3)*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)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(8*b^4*d^3*x^3 + 3*a*b^3*c^2*d - 100*a^2*b
^2*c*d^2 + 105*a^3*b*d^3 + 14*(b^4*c*d^2 - a*b^3*d^3)*x^2 + (3*b^4*c^2*d - 38*a*b^3*c*d^2 + 35*a^2*b^2*d^3)*x)
*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^2*x + a*b^5*d^2), 1/48*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 +
35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*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^4*
d^3*x^3 + 3*a*b^3*c^2*d - 100*a^2*b^2*c*d^2 + 105*a^3*b*d^3 + 14*(b^4*c*d^2 - a*b^3*d^3)*x^2 + (3*b^4*c^2*d -
38*a*b^3*c*d^2 + 35*a^2*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^2*x + a*b^5*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (c + d x\right )^{\frac{3}{2}}}{\left (a + b x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.52189, size = 468, normalized size = 1.9 \begin{align*} \frac{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{6}} + \frac{7 \, b^{18} c d^{4}{\left | b \right |} - 19 \, a b^{17} d^{5}{\left | b \right |}}{b^{23} d^{4}}\right )} + \frac{3 \,{\left (b^{19} c^{2} d^{3}{\left | b \right |} - 22 \, a b^{18} c d^{4}{\left | b \right |} + 29 \, a^{2} b^{17} d^{5}{\left | b \right |}\right )}}{b^{23} d^{4}}\right )} - \frac{4 \,{\left (\sqrt{b d} a^{2} b^{2} c^{2}{\left | b \right |} - 2 \, \sqrt{b d} a^{3} b c d{\left | b \right |} + \sqrt{b d} a^{4} d^{2}{\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^{5}} + \frac{{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} + 9 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} - 45 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + 35 \, \sqrt{b d} a^{3} d^{3}{\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 )}{16 \, b^{6} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*d*abs(b)/b^6 + (7*b^18*c*d^4*
abs(b) - 19*a*b^17*d^5*abs(b))/(b^23*d^4)) + 3*(b^19*c^2*d^3*abs(b) - 22*a*b^18*c*d^4*abs(b) + 29*a^2*b^17*d^5
*abs(b))/(b^23*d^4)) - 4*(sqrt(b*d)*a^2*b^2*c^2*abs(b) - 2*sqrt(b*d)*a^3*b*c*d*abs(b) + sqrt(b*d)*a^4*d^2*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^5) + 1/16*(sqrt(b*d
)*b^3*c^3*abs(b) + 9*sqrt(b*d)*a*b^2*c^2*d*abs(b) - 45*sqrt(b*d)*a^2*b*c*d^2*abs(b) + 35*sqrt(b*d)*a^3*d^3*abs
(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^6*d^2)