3.8.67 \(\int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=361 \[ \frac {7 a d+5 b c}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} c^{9/2}}+\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {d \sqrt {a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{12 a^3 c^3 (c+d x)^{3/2} (b c-a d)^2}+\frac {d \sqrt {a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{12 a^3 c^4 \sqrt {c+d x} (b c-a d)^3}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 361, normalized size of antiderivative = 1.00,
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 =
{103, 151, 152, 12, 93, 208} \begin {gather*} \frac {d \sqrt {a+b x} \left (-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4-30 a b^3 c^3 d+45 b^4 c^4\right )}{12 a^3 c^4 \sqrt {c+d x} (b c-a d)^3}+\frac {d \sqrt {a+b x} \left (-33 a^2 b c d^2+35 a^3 d^3-15 a b^2 c^2 d+45 b^3 c^3\right )}{12 a^3 c^3 (c+d x)^{3/2} (b c-a d)^2}+\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {5 \left (7 a^2 d^2+6 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^{7/2} c^{9/2}}+\frac {7 a d+5 b c}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(b*(15*b^2*c^2 - 7*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - 1/(2*a*c*x^2*Sqrt[a + b*x
]*(c + d*x)^(3/2)) + (5*b*c + 7*a*d)/(4*a^2*c^2*x*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (d*(45*b^3*c^3 - 15*a*b^2*c
^2*d - 33*a^2*b*c*d^2 + 35*a^3*d^3)*Sqrt[a + b*x])/(12*a^3*c^3*(b*c - a*d)^2*(c + d*x)^(3/2)) + (d*(45*b^4*c^4
- 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d^4)*Sqrt[a + b*x])/(12*a^3*c^4*(b*c - a*d)
^3*Sqrt[c + d*x]) - (5*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d
*x])])/(4*a^(7/2)*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 93
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 103
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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
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] && IntegerQ[m]
Rule 152
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin {align*} \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (5 b c+7 a d)+4 b d x}{x^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{2 a c}\\ &=-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\int \frac {\frac {5}{4} \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )+\frac {3}{2} b d (5 b c+7 a d) x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{2 a^2 c^2}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\int \frac {\frac {5}{8} (b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )+\frac {1}{2} b d \left (15 b^2 c^2-7 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a^3 c^2 (b c-a d)}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 \int \frac {-\frac {15}{16} (b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )-\frac {1}{8} b d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a^3 c^3 (b c-a d)^2}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {d \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^3 c^4 (b c-a d)^3 \sqrt {c+d x}}+\frac {4 \int \frac {15 (b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^3 c^4 (b c-a d)^3}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {d \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^3 c^4 (b c-a d)^3 \sqrt {c+d x}}+\frac {\left (5 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3 c^4}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {d \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^3 c^4 (b c-a d)^3 \sqrt {c+d x}}+\frac {\left (5 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^3 c^4}\\ &=\frac {b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{2 a c x^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 b c+7 a d}{4 a^2 c^2 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {d \left (45 b^3 c^3-15 a b^2 c^2 d-33 a^2 b c d^2+35 a^3 d^3\right ) \sqrt {a+b x}}{12 a^3 c^3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {d \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x}}{12 a^3 c^4 (b c-a d)^3 \sqrt {c+d x}}-\frac {5 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 374, normalized size = 1.04 \begin {gather*} \frac {3 a^{3/2} c^{5/2} x (b c-a d)^3 (7 a d+5 b c)+6 a^{5/2} c^{7/2} (a d-b c)^3-3 \sqrt {a} c^{5/2} x^2 (b c-a d)^2 \left (7 a^2 b d^2-15 b^3 c^2\right )+x^2 \left (-\sqrt {a} c^{3/2} d (a+b x) (a d-b c) \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )-\sqrt {a+b x} (c+d x) \left (15 \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c} d \sqrt {a+b x} \left (105 a^4 d^4-190 a^3 b c d^3+36 a^2 b^2 c^2 d^2+30 a b^3 c^3 d-45 b^4 c^4\right )\right )\right )}{12 a^{7/2} c^{9/2} x^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(6*a^(5/2)*c^(7/2)*(-(b*c) + a*d)^3 + 3*a^(3/2)*c^(5/2)*(b*c - a*d)^3*(5*b*c + 7*a*d)*x - 3*Sqrt[a]*c^(5/2)*(b
*c - a*d)^2*(-15*b^3*c^2 + 7*a^2*b*d^2)*x^2 + x^2*(-(Sqrt[a]*c^(3/2)*d*(-(b*c) + a*d)*(45*b^3*c^3 - 15*a*b^2*c
^2*d - 33*a^2*b*c*d^2 + 35*a^3*d^3)*(a + b*x)) - Sqrt[a + b*x]*(c + d*x)*(Sqrt[a]*Sqrt[c]*d*(-45*b^4*c^4 + 30*
a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 190*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x] + 15*(b*c - a*d)^3*(3*b^2*c^2
+ 6*a*b*c*d + 7*a^2*d^2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(12*a^(7/2)
*c^(9/2)*(b*c - a*d)^3*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2))
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IntegrateAlgebraic [A] time = 0.56, size = 512, normalized size = 1.42 \begin {gather*} \frac {(a+b x)^{3/2} \left (\frac {105 a^6 d^5 (c+d x)^3}{(a+b x)^3}-\frac {175 a^5 c d^5 (c+d x)^2}{(a+b x)^2}-\frac {225 a^5 b c d^4 (c+d x)^3}{(a+b x)^3}+\frac {90 a^4 b^2 c^2 d^3 (c+d x)^3}{(a+b x)^3}+\frac {56 a^4 c^2 d^5 (c+d x)}{a+b x}+\frac {375 a^4 b c^2 d^4 (c+d x)^2}{(a+b x)^2}+\frac {30 a^3 b^3 c^3 d^2 (c+d x)^3}{(a+b x)^3}-\frac {150 a^3 b^2 c^3 d^3 (c+d x)^2}{(a+b x)^2}-\frac {120 a^3 b c^3 d^4 (c+d x)}{a+b x}+8 a^3 c^3 d^5-\frac {24 a^2 b^5 c^4 (c+d x)^4}{(a+b x)^4}-\frac {75 a^2 b^4 c^4 d (c+d x)^3}{(a+b x)^3}+\frac {30 a^2 b^3 c^4 d^2 (c+d x)^2}{(a+b x)^2}-\frac {45 b^5 c^6 (c+d x)^2}{(a+b x)^2}+\frac {75 a b^5 c^5 (c+d x)^3}{(a+b x)^3}+\frac {45 a b^4 c^5 d (c+d x)^2}{(a+b x)^2}\right )}{12 a^3 c^4 (c+d x)^{3/2} (a d-b c)^3 \left (\frac {a (c+d x)}{a+b x}-c\right )^2}-\frac {5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{7/2} c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
((a + b*x)^(3/2)*(8*a^3*c^3*d^5 - (120*a^3*b*c^3*d^4*(c + d*x))/(a + b*x) + (56*a^4*c^2*d^5*(c + d*x))/(a + b*
x) - (45*b^5*c^6*(c + d*x)^2)/(a + b*x)^2 + (45*a*b^4*c^5*d*(c + d*x)^2)/(a + b*x)^2 + (30*a^2*b^3*c^4*d^2*(c
+ d*x)^2)/(a + b*x)^2 - (150*a^3*b^2*c^3*d^3*(c + d*x)^2)/(a + b*x)^2 + (375*a^4*b*c^2*d^4*(c + d*x)^2)/(a + b
*x)^2 - (175*a^5*c*d^5*(c + d*x)^2)/(a + b*x)^2 + (75*a*b^5*c^5*(c + d*x)^3)/(a + b*x)^3 - (75*a^2*b^4*c^4*d*(
c + d*x)^3)/(a + b*x)^3 + (30*a^3*b^3*c^3*d^2*(c + d*x)^3)/(a + b*x)^3 + (90*a^4*b^2*c^2*d^3*(c + d*x)^3)/(a +
b*x)^3 - (225*a^5*b*c*d^4*(c + d*x)^3)/(a + b*x)^3 + (105*a^6*d^5*(c + d*x)^3)/(a + b*x)^3 - (24*a^2*b^5*c^4*
(c + d*x)^4)/(a + b*x)^4))/(12*a^3*c^4*(-(b*c) + a*d)^3*(c + d*x)^(3/2)*(-c + (a*(c + d*x))/(a + b*x))^2) - (5
*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(4*a^(7/2)*c^(9
/2))
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fricas [B] time = 28.23, size = 2002, normalized size = 5.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
[1/48*(15*((3*b^6*c^5*d^2 - 3*a*b^5*c^4*d^3 - 2*a^2*b^4*c^3*d^4 - 6*a^3*b^3*c^2*d^5 + 15*a^4*b^2*c*d^6 - 7*a^5
*b*d^7)*x^5 + (6*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 7*a^2*b^4*c^4*d^3 - 14*a^3*b^3*c^3*d^4 + 24*a^4*b^2*c^2*d^5 + a
^5*b*c*d^6 - 7*a^6*d^7)*x^4 + (3*b^6*c^7 + 3*a*b^5*c^6*d - 8*a^2*b^4*c^5*d^2 - 10*a^3*b^3*c^4*d^3 + 3*a^4*b^2*
c^3*d^4 + 23*a^5*b*c^2*d^5 - 14*a^6*c*d^6)*x^3 + (3*a*b^5*c^7 - 3*a^2*b^4*c^6*d - 2*a^3*b^3*c^5*d^2 - 6*a^4*b^
2*c^4*d^3 + 15*a^5*b*c^3*d^4 - 7*a^6*c^2*d^5)*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
^3*b^3*c^7 - 18*a^4*b^2*c^6*d + 18*a^5*b*c^5*d^2 - 6*a^6*c^4*d^3 - (45*a*b^5*c^5*d^2 - 30*a^2*b^4*c^4*d^3 - 36
*a^3*b^3*c^3*d^4 + 190*a^4*b^2*c^2*d^5 - 105*a^5*b*c*d^6)*x^4 - (90*a*b^5*c^6*d - 45*a^2*b^4*c^5*d^2 - 84*a^3*
b^3*c^4*d^3 + 222*a^4*b^2*c^3*d^4 + 50*a^5*b*c^2*d^5 - 105*a^6*c*d^6)*x^3 - (45*a*b^5*c^7 - 66*a^3*b^3*c^5*d^2
- 12*a^4*b^2*c^4*d^3 + 237*a^5*b*c^3*d^4 - 140*a^6*c^2*d^5)*x^2 - 3*(5*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d - 6*a^4*
b^2*c^5*d^2 + 16*a^5*b*c^4*d^3 - 7*a^6*c^3*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^4*b^4*c^8*d^2 - 3*a^5*b^3*
c^7*d^3 + 3*a^6*b^2*c^6*d^4 - a^7*b*c^5*d^5)*x^5 + (2*a^4*b^4*c^9*d - 5*a^5*b^3*c^8*d^2 + 3*a^6*b^2*c^7*d^3 +
a^7*b*c^6*d^4 - a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 - a^5*b^3*c^9*d - 3*a^6*b^2*c^8*d^2 + 5*a^7*b*c^7*d^3 - 2*a^8
*c^6*d^4)*x^3 + (a^5*b^3*c^10 - 3*a^6*b^2*c^9*d + 3*a^7*b*c^8*d^2 - a^8*c^7*d^3)*x^2), 1/24*(15*((3*b^6*c^5*d^
2 - 3*a*b^5*c^4*d^3 - 2*a^2*b^4*c^3*d^4 - 6*a^3*b^3*c^2*d^5 + 15*a^4*b^2*c*d^6 - 7*a^5*b*d^7)*x^5 + (6*b^6*c^6
*d - 3*a*b^5*c^5*d^2 - 7*a^2*b^4*c^4*d^3 - 14*a^3*b^3*c^3*d^4 + 24*a^4*b^2*c^2*d^5 + a^5*b*c*d^6 - 7*a^6*d^7)*
x^4 + (3*b^6*c^7 + 3*a*b^5*c^6*d - 8*a^2*b^4*c^5*d^2 - 10*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 + 23*a^5*b*c^2*d
^5 - 14*a^6*c*d^6)*x^3 + (3*a*b^5*c^7 - 3*a^2*b^4*c^6*d - 2*a^3*b^3*c^5*d^2 - 6*a^4*b^2*c^4*d^3 + 15*a^5*b*c^3
*d^4 - 7*a^6*c^2*d^5)*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^3*b^3*c^7 - 18*a^4*b^2*c^6*d + 18*a^5*b*c^5*d^2 -
6*a^6*c^4*d^3 - (45*a*b^5*c^5*d^2 - 30*a^2*b^4*c^4*d^3 - 36*a^3*b^3*c^3*d^4 + 190*a^4*b^2*c^2*d^5 - 105*a^5*b*
c*d^6)*x^4 - (90*a*b^5*c^6*d - 45*a^2*b^4*c^5*d^2 - 84*a^3*b^3*c^4*d^3 + 222*a^4*b^2*c^3*d^4 + 50*a^5*b*c^2*d^
5 - 105*a^6*c*d^6)*x^3 - (45*a*b^5*c^7 - 66*a^3*b^3*c^5*d^2 - 12*a^4*b^2*c^4*d^3 + 237*a^5*b*c^3*d^4 - 140*a^6
*c^2*d^5)*x^2 - 3*(5*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d - 6*a^4*b^2*c^5*d^2 + 16*a^5*b*c^4*d^3 - 7*a^6*c^3*d^4)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/((a^4*b^4*c^8*d^2 - 3*a^5*b^3*c^7*d^3 + 3*a^6*b^2*c^6*d^4 - a^7*b*c^5*d^5)*x^5 +
(2*a^4*b^4*c^9*d - 5*a^5*b^3*c^8*d^2 + 3*a^6*b^2*c^7*d^3 + a^7*b*c^6*d^4 - a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 -
a^5*b^3*c^9*d - 3*a^6*b^2*c^8*d^2 + 5*a^7*b*c^7*d^3 - 2*a^8*c^6*d^4)*x^3 + (a^5*b^3*c^10 - 3*a^6*b^2*c^9*d + 3
*a^7*b*c^8*d^2 - a^8*c^7*d^3)*x^2)]
________________________________________________________________________________________
giac [B] time = 64.59, size = 1482, normalized size = 4.11
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x, algorithm="giac")
[Out]
4*sqrt(b*d)*b^6/((a^3*b^2*c^2*abs(b) - 2*a^4*b*c*d*abs(b) + a^5*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)) + 2/3*sqrt(b*x + a)*((14*b^6*c^7*d^6*abs(b) - 37*a*b^5*c^6*
d^7*abs(b) + 32*a^2*b^4*c^5*d^8*abs(b) - 9*a^3*b^3*c^4*d^9*abs(b))*(b*x + a)/(b^7*c^13*d - 5*a*b^6*c^12*d^2 +
10*a^2*b^5*c^11*d^3 - 10*a^3*b^4*c^10*d^4 + 5*a^4*b^3*c^9*d^5 - a^5*b^2*c^8*d^6) + 3*(5*b^7*c^8*d^5*abs(b) - 1
8*a*b^6*c^7*d^6*abs(b) + 24*a^2*b^5*c^6*d^7*abs(b) - 14*a^3*b^4*c^5*d^8*abs(b) + 3*a^4*b^3*c^4*d^9*abs(b))/(b^
7*c^13*d - 5*a*b^6*c^12*d^2 + 10*a^2*b^5*c^11*d^3 - 10*a^3*b^4*c^10*d^4 + 5*a^4*b^3*c^9*d^5 - a^5*b^2*c^8*d^6)
)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/4*(3*sqrt(b*d)*b^4*c^2 + 6*sqrt(b*d)*a*b^3*c*d + 7*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^3*b*c^4*abs(b)) + 1/2*(7*sqrt(b*d)*b^10*c^5 - 17*sqrt(b*d)*a*b^9*c^4*d - 2*sqr
t(b*d)*a^2*b^8*c^3*d^2 + 38*sqrt(b*d)*a^3*b^7*c^2*d^3 - 37*sqrt(b*d)*a^4*b^6*c*d^4 + 11*sqrt(b*d)*a^5*b^5*d^5
- 21*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 - 16*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 + 62*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 + 8*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^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 + 21*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 + 47*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 + 43*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)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*d^3 - 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^4*c^2 - 14*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^3*c^4*abs(b))
________________________________________________________________________________________
maple [B] time = 0.06, size = 2216, normalized size = 6.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)
[Out]
-1/24*(180*x^3*b^5*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-210*x^3*a^5*d^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/
2)+105*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^6*d^7-45*ln((a*d*x+b*c*x+2*a*c+2*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*b^6*c^7+90*x^2*b^5*c^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+12*a^5*
c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-12*a^2*b^3*c^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+105*ln((a*d*x+b
*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a^5*b*d^7-45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2))/x)*x^5*b^6*c^5*d^2-90*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4
*b^6*c^6*d+210*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a^6*c*d^6+105*ln((a*d*x+b*c
*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^6*c^2*d^5-45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2))/x)*x^2*a*b^5*c^7-280*x^2*a^5*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-42*x*a^5*c^2*d^4*(
(b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30*x*a*b^4*c^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-36*a^4*b*c^4*d^2*((b*x+a
)*(d*x+c))^(1/2)*(a*c)^(1/2)+36*a^3*b^2*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-225*ln((a*d*x+b*c*x+2*a*c+2*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a^4*b^2*c*d^6+90*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2))/x)*x^5*a^3*b^3*c^2*d^5+30*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a^2*
b^4*c^3*d^4+45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a*b^5*c^4*d^3-15*ln((a*d*x+
b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^5*b*c*d^6-360*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^4*b^2*c^2*d^5+210*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
))/x)*x^4*a^3*b^3*c^3*d^4+105*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^2*b^4*c^4*
d^3+45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a*b^5*c^5*d^2-345*ln((a*d*x+b*c*x+2
*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a^5*b*c^2*d^5-45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2))/x)*x^3*a^4*b^2*c^3*d^4+150*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*
x^3*a^3*b^3*c^4*d^3+120*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a^2*b^4*c^5*d^2-45
*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^3*a*b^5*c^6*d-225*ln((a*d*x+b*c*x+2*a*c+2*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^5*b*c^3*d^4+90*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2))/x)*x^2*a^4*b^2*c^4*d^3+30*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^3*b
^3*c^5*d^2+45*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^2*b^4*c^6*d-210*x^4*a^4*b*
d^6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+90*x^4*b^5*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+96*x*a^4*b*c^3*
d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-36*x*a^3*b^2*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-48*x*a^2*b^3*
c^5*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+380*x^4*a^3*b^2*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-72*x^4*a^2
*b^3*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-60*x^4*a*b^4*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+100*
x^3*a^4*b*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+444*x^3*a^3*b^2*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2
)-168*x^3*a^2*b^3*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-90*x^3*a*b^4*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*
c)^(1/2)+474*x^2*a^4*b*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-24*x^2*a^3*b^2*c^3*d^3*((b*x+a)*(d*x+c))^(1
/2)*(a*c)^(1/2)-132*x^2*a^2*b^3*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2))/c^4/a^3/(a*c)^(1/2)/x^2/(a*d-b*c)
^3/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(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^3), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,{\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^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x)
[Out]
int(1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \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**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
Integral(1/(x**3*(a + b*x)**(3/2)*(c + d*x)**(5/2)), x)
________________________________________________________________________________________