3.8.55 \(\int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [755]

Optimal. Leaf size=277 \[ \frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {d \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x}}{12 a^2 c^4 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{9/2}} \]

[Out]

-1/4*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(9/2)+1/
12*d*(-35*a^2*d^2+18*a*b*c*d+9*b^2*c^2)*(b*x+a)^(1/2)/a^2/c^3/(-a*d+b*c)/(d*x+c)^(3/2)-1/2*(b*x+a)^(1/2)/a/c/x
^2/(d*x+c)^(3/2)+1/4*(7*a*d+3*b*c)*(b*x+a)^(1/2)/a^2/c^2/x/(d*x+c)^(3/2)+1/12*d*(105*a^3*d^3-145*a^2*b*c*d^2+1
5*a*b^2*c^2*d+9*b^3*c^3)*(b*x+a)^(1/2)/a^2/c^4/(-a*d+b*c)^2/(d*x+c)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 277, normalized size of antiderivative = 1.00, 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 = {105, 156, 157, 12, 95, 214} \begin {gather*} \frac {d \sqrt {a+b x} \left (-35 a^2 d^2+18 a b c d+9 b^2 c^2\right )}{12 a^2 c^3 (c+d x)^{3/2} (b c-a d)}+\frac {\sqrt {a+b x} (7 a d+3 b c)}{4 a^2 c^2 x (c+d x)^{3/2}}-\frac {\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{9/2}}+\frac {d \sqrt {a+b x} \left (105 a^3 d^3-145 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{12 a^2 c^4 \sqrt {c+d x} (b c-a d)^2}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(d*(9*b^2*c^2 + 18*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x])/(12*a^2*c^3*(b*c - a*d)*(c + d*x)^(3/2)) - Sqrt[a + b*
x]/(2*a*c*x^2*(c + d*x)^(3/2)) + ((3*b*c + 7*a*d)*Sqrt[a + b*x])/(4*a^2*c^2*x*(c + d*x)^(3/2)) + (d*(9*b^3*c^3
+ 15*a*b^2*c^2*d - 145*a^2*b*c*d^2 + 105*a^3*d^3)*Sqrt[a + b*x])/(12*a^2*c^4*(b*c - a*d)^2*Sqrt[c + d*x]) - (
(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(
9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 214

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

Rubi steps

\begin {align*} \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{5/2}} \, dx &=-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (3 b c+7 a d)+3 b d x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c}\\ &=-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{4} \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )+b d (3 b c+7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a^2 c^2}\\ &=\frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}-\frac {\int \frac {-\frac {3}{8} (b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )-\frac {1}{4} b d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a^2 c^3 (b c-a d)}\\ &=\frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {d \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x}}{12 a^2 c^4 (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \int \frac {3 (b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^2 c^4 (b c-a d)^2}\\ &=\frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {d \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x}}{12 a^2 c^4 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^2 c^4}\\ &=\frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {d \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x}}{12 a^2 c^4 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^2 c^4}\\ &=\frac {d \left (9 b^2 c^2+18 a b c d-35 a^2 d^2\right ) \sqrt {a+b x}}{12 a^2 c^3 (b c-a d) (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{2 a c x^2 (c+d x)^{3/2}}+\frac {(3 b c+7 a d) \sqrt {a+b x}}{4 a^2 c^2 x (c+d x)^{3/2}}+\frac {d \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt {a+b x}}{12 a^2 c^4 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 224, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x} \left (9 b^3 c^3 x (c+d x)^2-3 a b^2 c^2 (2 c-5 d x) (c+d x)^2+a^2 b c d \left (12 c^3-33 c^2 d x-198 c d^2 x^2-145 d^3 x^3\right )+a^3 d^2 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )}{12 a^2 c^4 (b c-a d)^2 x^2 (c+d x)^{3/2}}-\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*(9*b^3*c^3*x*(c + d*x)^2 - 3*a*b^2*c^2*(2*c - 5*d*x)*(c + d*x)^2 + a^2*b*c*d*(12*c^3 - 33*c^2*d
*x - 198*c*d^2*x^2 - 145*d^3*x^3) + a^3*d^2*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3)))/(12*a^2*c^4*
(b*c - a*d)^2*x^2*(c + d*x)^(3/2)) - ((3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1287\) vs. \(2(239)=478\).
time = 0.08, size = 1288, normalized size = 4.65

method	result	size
default	\(\text {Expression too large to display}\)	\(1288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*(b*x+a)^(1/2)/a^2/c^4*(-24*a^2*b*c^4*d*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-180*ln((a*d*x+b*c*x+2*(a*c)^(
1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b*c*d^5*x^4+54*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2
)+2*a*c)/x)*a^2*b^2*c^2*d^4*x^4+12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^3*c^3*d
^3*x^4+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*d^6*x^4+9*ln((a*d*x+b*c*x+2*(a*
c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^4*c^6*x^2-360*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)+2*a*c)/x)*a^3*b*c^2*d^4*x^3+108*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^2*c^3
*d^3*x^3+24*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a*b^3*c^4*d^2*x^3-180*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*b*c^3*d^3*x^2+54*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c
)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b^2*c^4*d^2*x^2+12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)
/x)*a*b^3*c^5*d*x^2-36*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^3*c^4*d*x^2-42*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2
)*a^3*c^2*d^3*x-18*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^3*c^3*d^2*x^3-280*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)
*a^3*c*d^4*x^2-210*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^3*d^5*x^3-18*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*b^3*
c^5*x+210*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*c*d^5*x^3+105*ln((a*d*x+b*c*x+2*
(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^4*c^2*d^4*x^2+12*a^3*c^3*d^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)+12*a*b^2*c^5*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*
a*c)/x)*b^4*c^4*d^2*x^4+18*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*b^4*c^5*d*x^3+290*(
(d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b*c*d^4*x^3-30*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b^2*c^2*d^3*x^3+39
6*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d^3*x^2-48*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b^2*c^3*d^2*x
^2+66*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a^2*b*c^3*d^2*x-6*((d*x+c)*(b*x+a))^(1/2)*(a*c)^(1/2)*a*b^2*c^4*d*x)
/(a*c)^(1/2)/x^2/(a*d-b*c)^2/((d*x+c)*(b*x+a))^(1/2)/(d*x+c)^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (239) = 478\).
time = 2.92, size = 1186, normalized size = 4.28 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, b^{4} c^{4} d^{2} + 4 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} - 60 \, a^{3} b c d^{5} + 35 \, a^{4} d^{6}\right )} x^{4} + 2 \, {\left (3 \, b^{4} c^{5} d + 4 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} - 60 \, a^{3} b c^{2} d^{4} + 35 \, a^{4} c d^{5}\right )} x^{3} + {\left (3 \, b^{4} c^{6} + 4 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} - 60 \, a^{3} b c^{3} d^{3} + 35 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{2} b^{2} c^{6} - 12 \, a^{3} b c^{5} d + 6 \, a^{4} c^{4} d^{2} - {\left (9 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} - 145 \, a^{3} b c^{2} d^{4} + 105 \, a^{4} c d^{5}\right )} x^{3} - 2 \, {\left (9 \, a b^{3} c^{5} d + 12 \, a^{2} b^{2} c^{4} d^{2} - 99 \, a^{3} b c^{3} d^{3} + 70 \, a^{4} c^{2} d^{4}\right )} x^{2} - 3 \, {\left (3 \, a b^{3} c^{6} + a^{2} b^{2} c^{5} d - 11 \, a^{3} b c^{4} d^{2} + 7 \, a^{4} c^{3} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left ({\left (a^{3} b^{2} c^{7} d^{2} - 2 \, a^{4} b c^{6} d^{3} + a^{5} c^{5} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{2} c^{8} d - 2 \, a^{4} b c^{7} d^{2} + a^{5} c^{6} d^{3}\right )} x^{3} + {\left (a^{3} b^{2} c^{9} - 2 \, a^{4} b c^{8} d + a^{5} c^{7} d^{2}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{4} c^{4} d^{2} + 4 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} - 60 \, a^{3} b c d^{5} + 35 \, a^{4} d^{6}\right )} x^{4} + 2 \, {\left (3 \, b^{4} c^{5} d + 4 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} - 60 \, a^{3} b c^{2} d^{4} + 35 \, a^{4} c d^{5}\right )} x^{3} + {\left (3 \, b^{4} c^{6} + 4 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} - 60 \, a^{3} b c^{3} d^{3} + 35 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{2} b^{2} c^{6} - 12 \, a^{3} b c^{5} d + 6 \, a^{4} c^{4} d^{2} - {\left (9 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} - 145 \, a^{3} b c^{2} d^{4} + 105 \, a^{4} c d^{5}\right )} x^{3} - 2 \, {\left (9 \, a b^{3} c^{5} d + 12 \, a^{2} b^{2} c^{4} d^{2} - 99 \, a^{3} b c^{3} d^{3} + 70 \, a^{4} c^{2} d^{4}\right )} x^{2} - 3 \, {\left (3 \, a b^{3} c^{6} + a^{2} b^{2} c^{5} d - 11 \, a^{3} b c^{4} d^{2} + 7 \, a^{4} c^{3} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left ({\left (a^{3} b^{2} c^{7} d^{2} - 2 \, a^{4} b c^{6} d^{3} + a^{5} c^{5} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{2} c^{8} d - 2 \, a^{4} b c^{7} d^{2} + a^{5} c^{6} d^{3}\right )} x^{3} + {\left (a^{3} b^{2} c^{9} - 2 \, a^{4} b c^{8} d + a^{5} c^{7} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/48*(3*((3*b^4*c^4*d^2 + 4*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d^4 - 60*a^3*b*c*d^5 + 35*a^4*d^6)*x^4 + 2*(3*b^4*
c^5*d + 4*a*b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 - 60*a^3*b*c^2*d^4 + 35*a^4*c*d^5)*x^3 + (3*b^4*c^6 + 4*a*b^3*c^5
*d + 18*a^2*b^2*c^4*d^2 - 60*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^2)*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*
c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x
)/x^2) - 4*(6*a^2*b^2*c^6 - 12*a^3*b*c^5*d + 6*a^4*c^4*d^2 - (9*a*b^3*c^4*d^2 + 15*a^2*b^2*c^3*d^3 - 145*a^3*b
*c^2*d^4 + 105*a^4*c*d^5)*x^3 - 2*(9*a*b^3*c^5*d + 12*a^2*b^2*c^4*d^2 - 99*a^3*b*c^3*d^3 + 70*a^4*c^2*d^4)*x^2
- 3*(3*a*b^3*c^6 + a^2*b^2*c^5*d - 11*a^3*b*c^4*d^2 + 7*a^4*c^3*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^
2*c^7*d^2 - 2*a^4*b*c^6*d^3 + a^5*c^5*d^4)*x^4 + 2*(a^3*b^2*c^8*d - 2*a^4*b*c^7*d^2 + a^5*c^6*d^3)*x^3 + (a^3*
b^2*c^9 - 2*a^4*b*c^8*d + a^5*c^7*d^2)*x^2), 1/24*(3*((3*b^4*c^4*d^2 + 4*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d^4 -
60*a^3*b*c*d^5 + 35*a^4*d^6)*x^4 + 2*(3*b^4*c^5*d + 4*a*b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 - 60*a^3*b*c^2*d^4 +
35*a^4*c*d^5)*x^3 + (3*b^4*c^6 + 4*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 - 60*a^3*b*c^3*d^3 + 35*a^4*c^2*d^4)*x^2)*
sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 +
(a*b*c^2 + a^2*c*d)*x)) - 2*(6*a^2*b^2*c^6 - 12*a^3*b*c^5*d + 6*a^4*c^4*d^2 - (9*a*b^3*c^4*d^2 + 15*a^2*b^2*c^
3*d^3 - 145*a^3*b*c^2*d^4 + 105*a^4*c*d^5)*x^3 - 2*(9*a*b^3*c^5*d + 12*a^2*b^2*c^4*d^2 - 99*a^3*b*c^3*d^3 + 70
*a^4*c^2*d^4)*x^2 - 3*(3*a*b^3*c^6 + a^2*b^2*c^5*d - 11*a^3*b*c^4*d^2 + 7*a^4*c^3*d^3)*x)*sqrt(b*x + a)*sqrt(d
*x + c))/((a^3*b^2*c^7*d^2 - 2*a^4*b*c^6*d^3 + a^5*c^5*d^4)*x^4 + 2*(a^3*b^2*c^8*d - 2*a^4*b*c^7*d^2 + a^5*c^6
*d^3)*x^3 + (a^3*b^2*c^9 - 2*a^4*b*c^8*d + a^5*c^7*d^2)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(a + b*x)*(c + d*x)**(5/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1234 vs. \(2 (239) = 478\).
time = 1.77, size = 1234, normalized size = 4.45 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (11 \, b^{4} c^{5} d^{5} {\left | b \right |} - 9 \, a b^{3} c^{4} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{10} d - 2 \, a b^{3} c^{9} d^{2} + a^{2} b^{2} c^{8} d^{3}} + \frac {3 \, {\left (4 \, b^{5} c^{6} d^{4} {\left | b \right |} - 7 \, a b^{4} c^{5} d^{5} {\left | b \right |} + 3 \, a^{2} b^{3} c^{4} d^{6} {\left | b \right |}\right )}}{b^{4} c^{10} d - 2 \, a b^{3} c^{9} d^{2} + a^{2} b^{2} c^{8} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (3 \, \sqrt {b d} b^{4} c^{2} + 10 \, \sqrt {b d} a b^{3} c d + 35 \, \sqrt {b d} a^{2} b^{2} d^{2}\right )} \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{4 \, \sqrt {-a b c d} a^{2} b c^{4} {\left | b \right |}} + \frac {3 \, \sqrt {b d} b^{10} c^{5} - \sqrt {b d} a b^{9} c^{4} d - 26 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} + 54 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} - 41 \, \sqrt {b d} a^{4} b^{6} c d^{4} + 11 \, \sqrt {b d} a^{5} b^{5} d^{5} - 9 \, \sqrt {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} b^{8} c^{4} - 28 \, \sqrt {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} a b^{7} c^{3} d + 50 \, \sqrt {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} a^{2} b^{6} c^{2} d^{2} + 20 \, \sqrt {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} a^{3} b^{5} c d^{3} - 33 \, \sqrt {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} a^{4} b^{4} d^{4} + 9 \, \sqrt {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 )}^{4} b^{6} c^{3} + 39 \, \sqrt {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 )}^{4} a b^{5} c^{2} d + 31 \, \sqrt {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 )}^{4} a^{2} b^{4} c d^{2} + 33 \, \sqrt {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 )}^{4} a^{3} b^{3} d^{3} - 3 \, \sqrt {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 )}^{6} b^{4} c^{2} - 10 \, \sqrt {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 )}^{6} a b^{3} c d - 11 \, \sqrt {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 )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}\right )}^{2} a^{2} c^{4} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-2/3*sqrt(b*x + a)*((11*b^4*c^5*d^5*abs(b) - 9*a*b^3*c^4*d^6*abs(b))*(b*x + a)/(b^4*c^10*d - 2*a*b^3*c^9*d^2 +
a^2*b^2*c^8*d^3) + 3*(4*b^5*c^6*d^4*abs(b) - 7*a*b^4*c^5*d^5*abs(b) + 3*a^2*b^3*c^4*d^6*abs(b))/(b^4*c^10*d -
2*a*b^3*c^9*d^2 + a^2*b^2*c^8*d^3))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 1/4*(3*sqrt(b*d)*b^4*c^2 + 10*sqr
t(b*d)*a*b^3*c*d + 35*sqrt(b*d)*a^2*b^2*d^2)*arctan(-1/2*(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)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^2*b*c^4*abs(b)) + 1/2*(3*sqrt(b*d)*b^10*c
^5 - sqrt(b*d)*a*b^9*c^4*d - 26*sqrt(b*d)*a^2*b^8*c^3*d^2 + 54*sqrt(b*d)*a^3*b^7*c^2*d^3 - 41*sqrt(b*d)*a^4*b^
6*c*d^4 + 11*sqrt(b*d)*a^5*b^5*d^5 - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2*b^8*c^4 - 28*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^3*d + 50
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^6*c^2*d^2 + 20*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*c*d^3 - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^4*d^4 + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3 + 39*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^4*a*b^5*c^2*d + 31*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^4
*c*d^2 + 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*d^3 - 3*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*c^2 - 10*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^3*c*d - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^6*a^2*b^2*d^2)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^2*a^2*c^4*abs(b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x)^(1/2)*(c + d*x)^(5/2)),x)

[Out]

int(1/(x^3*(a + b*x)^(1/2)*(c + d*x)^(5/2)), x)

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