3.8.91 \(\int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\) [791]

Optimal. Leaf size=264 \[ -\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}} \]

[Out]

(5*a*d+3*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(7/2)-b*(-a*d+3*b*c)/a^2/c/(-a*d+
b*c)/(d*x+c)^(3/2)/(b*x+a)^(1/2)-1/a/c/x/(d*x+c)^(3/2)/(b*x+a)^(1/2)-1/3*d*(5*a^2*d^2-6*a*b*c*d+9*b^2*c^2)*(b*
x+a)^(1/2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(3/2)-1/3*d*(-15*a^3*d^3+31*a^2*b*c*d^2-9*a*b^2*c^2*d+9*b^3*c^3)*(b*x+
a)^(1/2)/a^2/c^3/(-a*d+b*c)^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 157, 12, 95, 214} \begin {gather*} \frac {(5 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac {d \sqrt {a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {d \sqrt {a+b x} \left (-15 a^3 d^3+31 a^2 b c d^2-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt {c+d x} (b c-a d)^3}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))) - 1/(a*c*x*Sqrt[a + b*x]*(c + d*x)^(3/2
)) - (d*(9*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (d*(9*b
^3*c^3 - 9*a*b^2*c^2*d + 31*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x])/(3*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) +
((3*b*c + 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(5/2)*c^(7/2))

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 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^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (3 b c+5 a d)+3 b d x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \int \frac {\frac {1}{4} (b c-a d) (3 b c+5 a d)+b d (3 b c-a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 \int \frac {-\frac {3}{8} (b c-a d)^2 (3 b c+5 a d)-\frac {1}{4} b d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {8 \int \frac {3 (b c-a d)^3 (3 b c+5 a d)}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(3 b c+5 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2 c^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(3 b c+5 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^2 c^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 235, normalized size = 0.89 \begin {gather*} \frac {9 b^4 c^3 x (c+d x)^2+3 a b^3 c^2 (c-3 d x) (c+d x)^2-a^4 d^3 \left (3 c^2+20 c d x+15 d^2 x^2\right )+a^3 b d^2 \left (9 c^3+39 c^2 d x+11 c d^2 x^2-15 d^3 x^3\right )+a^2 b^2 c d \left (-9 c^3-9 c^2 d x+33 c d^2 x^2+31 d^3 x^3\right )}{3 a^2 c^3 (-b c+a d)^3 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b^4*c^3*x*(c + d*x)^2 + 3*a*b^3*c^2*(c - 3*d*x)*(c + d*x)^2 - a^4*d^3*(3*c^2 + 20*c*d*x + 15*d^2*x^2) + a^3
*b*d^2*(9*c^3 + 39*c^2*d*x + 11*c*d^2*x^2 - 15*d^3*x^3) + a^2*b^2*c*d*(-9*c^3 - 9*c^2*d*x + 33*c*d^2*x^2 + 31*
d^3*x^3))/(3*a^2*c^3*(-(b*c) + a*d)^3*x*Sqrt[a + b*x]*(c + d*x)^(3/2)) + ((3*b*c + 5*a*d)*ArcTanh[(Sqrt[a]*Sqr
t[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(a^(5/2)*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1691\) vs. \(2(232)=464\).
time = 0.07, size = 1692, normalized size = 6.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1692\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-57*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*b*c^2*d^4*x^2+42*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^3*c^4*d^2*x^2-6*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^4*c^5*d*x^2-36*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
*b*c^3*d^3*x+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)*a^3*b^2*c^4*d^2*x+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^2*b^3*c^5*d*x-30*a^3*b*d^5*x^3*(a*c)^(1/2)*((d*x+c)*(
b*x+a))^(1/2)+18*b^4*c^3*d^2*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+36*b^4*c^4*d*x^2*(a*c)^(1/2)*((d*x+c)*(b*
x+a))^(1/2)-40*a^4*c*d^4*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+18*a^3*b*c^3*d^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^
(1/2)-18*a^2*b^2*c^4*d*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-36*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^2*c*d^5*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)*a^2*b^3*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^4*c^3*d^3*x^4-6*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*b*c*d^5*x^3-54*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^3*b^2*c^2*d^4*x^3+48*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^3*c^3*d^3*x^3+15*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^4*c^4*d^2*x^3+15*
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^5*d^6*x^3-9*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^5*c^6*x^2+62*a^2*b^2*c*d^4*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-18*a*b^3*
c^2*d^3*x^3*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+22*a^3*b*c*d^4*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+66*a^2*
b^2*c^2*d^3*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-30*a*b^3*c^3*d^2*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+7
8*a^3*b*c^2*d^3*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-18*a^2*b^2*c^3*d^2*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)
-6*a*b^3*c^4*d*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+15*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^5*c^2*d^4*x-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)*a*b^4*c^6*x-30*a^4*d
^5*x^2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+18*b^4*c^5*x*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-6*a^4*c^2*d^3*(a*c
)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+6*a*b^3*c^5*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+15*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*b*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^5*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^5*c^5*d*x^3+30*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^5*c*d^5*x^2)/a^2/c^3/(a*c)^(1/2)/x/(a*d-b*c)^3/
((d*x+c)*(b*x+a))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (232) = 464\).
time = 3.55, size = 1658, normalized size = 6.28 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} - 6 \, a^{2} b^{3} c^{2} d^{4} + 12 \, a^{3} b^{2} c d^{5} - 5 \, a^{4} b d^{6}\right )} x^{4} + {\left (6 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - 16 \, a^{2} b^{3} c^{3} d^{3} + 18 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 5 \, a^{5} d^{6}\right )} x^{3} + {\left (3 \, b^{5} c^{6} + 2 \, a b^{4} c^{5} d - 14 \, a^{2} b^{3} c^{4} d^{2} + 19 \, a^{4} b c^{2} d^{4} - 10 \, a^{5} c d^{5}\right )} x^{2} + {\left (3 \, a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 12 \, a^{4} b c^{3} d^{3} - 5 \, a^{5} c^{2} d^{4}\right )} x\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 (3 \, a^{2} b^{3} c^{6} - 9 \, a^{3} b^{2} c^{5} d + 9 \, a^{4} b c^{4} d^{2} - 3 \, a^{5} c^{3} d^{3} + {\left (9 \, a b^{4} c^{4} d^{2} - 9 \, a^{2} b^{3} c^{3} d^{3} + 31 \, a^{3} b^{2} c^{2} d^{4} - 15 \, a^{4} b c d^{5}\right )} x^{3} + {\left (18 \, a b^{4} c^{5} d - 15 \, a^{2} b^{3} c^{4} d^{2} + 33 \, a^{3} b^{2} c^{3} d^{3} + 11 \, a^{4} b c^{2} d^{4} - 15 \, a^{5} c d^{5}\right )} x^{2} + {\left (9 \, a b^{4} c^{6} - 3 \, a^{2} b^{3} c^{5} d - 9 \, a^{3} b^{2} c^{4} d^{2} + 39 \, a^{4} b c^{3} d^{3} - 20 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{4} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{3} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x\right )}}, -\frac {3 \, {\left ({\left (3 \, b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} - 6 \, a^{2} b^{3} c^{2} d^{4} + 12 \, a^{3} b^{2} c d^{5} - 5 \, a^{4} b d^{6}\right )} x^{4} + {\left (6 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - 16 \, a^{2} b^{3} c^{3} d^{3} + 18 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 5 \, a^{5} d^{6}\right )} x^{3} + {\left (3 \, b^{5} c^{6} + 2 \, a b^{4} c^{5} d - 14 \, a^{2} b^{3} c^{4} d^{2} + 19 \, a^{4} b c^{2} d^{4} - 10 \, a^{5} c d^{5}\right )} x^{2} + {\left (3 \, a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 12 \, a^{4} b c^{3} d^{3} - 5 \, a^{5} c^{2} d^{4}\right )} x\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 (3 \, a^{2} b^{3} c^{6} - 9 \, a^{3} b^{2} c^{5} d + 9 \, a^{4} b c^{4} d^{2} - 3 \, a^{5} c^{3} d^{3} + {\left (9 \, a b^{4} c^{4} d^{2} - 9 \, a^{2} b^{3} c^{3} d^{3} + 31 \, a^{3} b^{2} c^{2} d^{4} - 15 \, a^{4} b c d^{5}\right )} x^{3} + {\left (18 \, a b^{4} c^{5} d - 15 \, a^{2} b^{3} c^{4} d^{2} + 33 \, a^{3} b^{2} c^{3} d^{3} + 11 \, a^{4} b c^{2} d^{4} - 15 \, a^{5} c d^{5}\right )} x^{2} + {\left (9 \, a b^{4} c^{6} - 3 \, a^{2} b^{3} c^{5} d - 9 \, a^{3} b^{2} c^{4} d^{2} + 39 \, a^{4} b c^{3} d^{3} - 20 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{4} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{3} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 - 6*a^2*b^3*c^2*d^4 + 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*
c^5*d - 5*a*b^4*c^4*d^2 - 16*a^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (3*b^5*c^
6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^2 + (3*a*b^4*c^6 - 4*a^2*b^3*c^5*d
- 6*a^3*b^2*c^4*d^2 + 12*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*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*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + (9*a*b^4*c^4*d^2 - 9*a^2*b^3*c^3*d^3
+ 31*a^3*b^2*c^2*d^4 - 15*a^4*b*c*d^5)*x^3 + (18*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 33*a^3*b^2*c^3*d^3 + 11*a
^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (9*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 39*a^4*b*c^3*d^3 - 20*
a^5*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b
*c^4*d^5)*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^3 +
(a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (a^4*b^3*c^9 - 3*a^5
*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x), -1/6*(3*((3*b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 - 6*a^2*b^3*c^2*d^4
+ 12*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^4 + (6*b^5*c^5*d - 5*a*b^4*c^4*d^2 - 16*a^2*b^3*c^3*d^3 + 18*a^3*b^2*c^2*d
^4 + 2*a^4*b*c*d^5 - 5*a^5*d^6)*x^3 + (3*b^5*c^6 + 2*a*b^4*c^5*d - 14*a^2*b^3*c^4*d^2 + 19*a^4*b*c^2*d^4 - 10*
a^5*c*d^5)*x^2 + (3*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 12*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x)*sqr
t(-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*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + (9*a*b^4*c^4*d^2
- 9*a^2*b^3*c^3*d^3 + 31*a^3*b^2*c^2*d^4 - 15*a^4*b*c*d^5)*x^3 + (18*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 33*a^
3*b^2*c^3*d^3 + 11*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^2 + (9*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 39
*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5
*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^4 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 -
a^7*c^4*d^5)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 +
(a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x)]

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

[Out]

Integral(1/(x**2*(a + b*x)**(3/2)*(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. 1336 vs. \(2 (232) = 464\).
time = 3.40, size = 1336, normalized size = 5.06 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (11 \, b^{6} c^{6} d^{5} {\left | b \right |} - 28 \, a b^{5} c^{5} d^{6} {\left | b \right |} + 23 \, a^{2} b^{4} c^{4} d^{7} {\left | b \right |} - 6 \, a^{3} b^{3} c^{3} d^{8} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{11} d - 5 \, a b^{6} c^{10} d^{2} + 10 \, a^{2} b^{5} c^{9} d^{3} - 10 \, a^{3} b^{4} c^{8} d^{4} + 5 \, a^{4} b^{3} c^{7} d^{5} - a^{5} b^{2} c^{6} d^{6}} + \frac {6 \, {\left (2 \, b^{7} c^{7} d^{4} {\left | b \right |} - 7 \, a b^{6} c^{6} d^{5} {\left | b \right |} + 9 \, a^{2} b^{5} c^{5} d^{6} {\left | b \right |} - 5 \, a^{3} b^{4} c^{4} d^{7} {\left | b \right |} + a^{4} b^{3} c^{3} d^{8} {\left | b \right |}\right )}}{b^{7} c^{11} d - 5 \, a b^{6} c^{10} d^{2} + 10 \, a^{2} b^{5} c^{9} d^{3} - 10 \, a^{3} b^{4} c^{8} d^{4} + 5 \, a^{4} b^{3} c^{7} d^{5} - a^{5} b^{2} c^{6} d^{6}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{9} c^{5} - 9 \, \sqrt {b d} a b^{8} c^{4} d + 12 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} - 10 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} + 5 \, \sqrt {b d} a^{4} b^{5} c d^{4} - \sqrt {b d} a^{5} b^{4} d^{5} - 6 \, \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^{7} c^{4} + 2 \, \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^{6} c^{3} d - 6 \, \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^{5} c^{2} d^{2} + 2 \, \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^{4} c 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 )}^{4} b^{5} c^{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 )}^{4} a b^{4} c^{2} d - \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^{3} c d^{2} + \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^{2} d^{3}\right )}}{{\left (a^{2} b^{2} c^{5} {\left | b \right |} - 2 \, a^{3} b c^{4} d {\left | b \right |} + a^{4} c^{3} d^{2} {\left | b \right |}\right )} {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{4} c^{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} a b^{3} c 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^{2} d^{2} + 3 \, {\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^{2} c + {\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 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}\right )}} + \frac {{\left (3 \, \sqrt {b d} b^{3} c + 5 \, \sqrt {b d} a b^{2} d\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 )}{\sqrt {-a b c d} a^{2} b c^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x + a)*((11*b^6*c^6*d^5*abs(b) - 28*a*b^5*c^5*d^6*abs(b) + 23*a^2*b^4*c^4*d^7*abs(b) - 6*a^3*b^3*c
^3*d^8*abs(b))*(b*x + a)/(b^7*c^11*d - 5*a*b^6*c^10*d^2 + 10*a^2*b^5*c^9*d^3 - 10*a^3*b^4*c^8*d^4 + 5*a^4*b^3*
c^7*d^5 - a^5*b^2*c^6*d^6) + 6*(2*b^7*c^7*d^4*abs(b) - 7*a*b^6*c^6*d^5*abs(b) + 9*a^2*b^5*c^5*d^6*abs(b) - 5*a
^3*b^4*c^4*d^7*abs(b) + a^4*b^3*c^3*d^8*abs(b))/(b^7*c^11*d - 5*a*b^6*c^10*d^2 + 10*a^2*b^5*c^9*d^3 - 10*a^3*b
^4*c^8*d^4 + 5*a^4*b^3*c^7*d^5 - a^5*b^2*c^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 2*(3*sqrt(b*d)*b^9*
c^5 - 9*sqrt(b*d)*a*b^8*c^4*d + 12*sqrt(b*d)*a^2*b^7*c^3*d^2 - 10*sqrt(b*d)*a^3*b^6*c^2*d^3 + 5*sqrt(b*d)*a^4*
b^5*c*d^4 - sqrt(b*d)*a^5*b^4*d^5 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^2*b^7*c^4 + 2*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^6*c^3*d - 6*sq
rt(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^5*c^2*d^2 + 2*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^4*c*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))^4*b^5*c^3 - 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^4*c^2*d - 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^3*c*d^2 + 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^2*d^3)/((a^2
*b^2*c^5*abs(b) - 2*a^3*b*c^4*d*abs(b) + a^4*c^3*d^2*abs(b))*(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*
b^3*d^3 - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^4*c^2 + 2*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3*c*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^2*a^2*b^2*d^2 + 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c + (sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^6)) + (3*sqrt(b*d)*b^3*c + 5*sqrt(b*d)*a*b^2*d)*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^3*abs(b))

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

[Out]

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

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