3.766 \(\int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx\)
Optimal. Leaf size=241 \[ -\frac {\sqrt {c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt {a+b x}}+\frac {7 \sqrt {c+d x} (b c-a d)}{12 a^2 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac {b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}} \]
[Out]
1/8*(-a*d+b*c)*(-a^2*d^2-10*a*b*c*d+35*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c
^(3/2)-1/24*b*(3*a^2*d^2-100*a*b*c*d+105*b^2*c^2)*(d*x+c)^(1/2)/a^4/c/(b*x+a)^(1/2)-1/3*c*(d*x+c)^(1/2)/a/x^3/
(b*x+a)^(1/2)+7/12*(-a*d+b*c)*(d*x+c)^(1/2)/a^2/x^2/(b*x+a)^(1/2)-1/24*(-3*a*d+35*b*c)*(-a*d+b*c)*(d*x+c)^(1/2
)/a^3/c/x/(b*x+a)^(1/2)
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Rubi [A] time = 0.23, antiderivative size = 241, 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 =
{98, 151, 152, 12, 93, 208} \[ -\frac {b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt {a+b x}}+\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}+\frac {7 \sqrt {c+d x} (b c-a d)}{12 a^2 x^2 \sqrt {a+b x}}-\frac {\sqrt {c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)),x]
[Out]
-(b*(105*b^2*c^2 - 100*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x])/(24*a^4*c*Sqrt[a + b*x]) - (c*Sqrt[c + d*x])/(3*a*x
^3*Sqrt[a + b*x]) + (7*(b*c - a*d)*Sqrt[c + d*x])/(12*a^2*x^2*Sqrt[a + b*x]) - ((35*b*c - 3*a*d)*(b*c - a*d)*S
qrt[c + d*x])/(24*a^3*c*x*Sqrt[a + b*x]) + ((b*c - a*d)*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(9/2)*c^(3/2))
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 98
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])
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 {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}-\frac {\int \frac {\frac {7}{2} c (b c-a d)+3 d (b c-a d) x}{x^3 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 a}\\ &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}+\frac {\int \frac {\frac {1}{4} c (35 b c-3 a d) (b c-a d)+7 b c d (b c-a d) x}{x^2 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{6 a^2 c}\\ &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\int \frac {\frac {3}{8} c (b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )+\frac {1}{4} b c d (35 b c-3 a d) (b c-a d) x}{x (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\int \frac {3 c (b c-a d)^2 \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^4 c^2 (b c-a d)}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^4 c}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-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 )}{8 a^4 c}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}+\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 190, normalized size = 0.79 \[ \frac {\left (a^3 d^3+9 a^2 b c d^2-45 a b^2 c^2 d+35 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac {\sqrt {c+d x} \left (a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )+a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+5 a b^2 c x^2 (7 c-20 d x)+105 b^3 c^2 x^3\right )}{24 a^4 c x^3 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)),x]
[Out]
-1/24*(Sqrt[c + d*x]*(105*b^3*c^2*x^3 + 5*a*b^2*c*x^2*(7*c - 20*d*x) + a^2*b*x*(-14*c^2 - 38*c*d*x + 3*d^2*x^2
) + a^3*(8*c^2 + 14*c*d*x + 3*d^2*x^2)))/(a^4*c*x^3*Sqrt[a + b*x]) + ((35*b^3*c^3 - 45*a*b^2*c^2*d + 9*a^2*b*c
*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(9/2)*c^(3/2))
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fricas [A] time = 3.73, size = 638, normalized size = 2.65 \[ \left [\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}, -\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)/x^4/(b*x+a)^(3/2),x, algorithm="fricas")
[Out]
[1/96*(3*((35*b^4*c^3 - 45*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 + a^3*b*d^3)*x^4 + (35*a*b^3*c^3 - 45*a^2*b^2*c^2*d +
9*a^3*b*c*d^2 + a^4*d^3)*x^3)*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*(8*a^4*c^3 + (105*a*b^3*
c^3 - 100*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x^3 + (35*a^2*b^2*c^3 - 38*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 14*(a^3*b
*c^3 - a^4*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*b*c^2*x^4 + a^6*c^2*x^3), -1/48*(3*((35*b^4*c^3 - 45*a*
b^3*c^2*d + 9*a^2*b^2*c*d^2 + a^3*b*d^3)*x^4 + (35*a*b^3*c^3 - 45*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 + a^4*d^3)*x^3
)*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*(8*a^4*c^3 + (105*a*b^3*c^3 - 100*a^2*b^2*c^2*d + 3*a^3*b*c*d^2)*x^3 + (35*a^2*b
^2*c^3 - 38*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 14*(a^3*b*c^3 - a^4*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*b
*c^2*x^4 + a^6*c^2*x^3)]
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giac [B] time = 77.96, size = 2310, normalized size = 9.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)/x^4/(b*x+a)^(3/2),x, algorithm="giac")
[Out]
-4*(sqrt(b*d)*b^3*c^2*abs(b) - 2*sqrt(b*d)*a*b^2*c*d*abs(b) + sqrt(b*d)*a^2*b*d^2*abs(b))/((b^2*c - a*b*d - (s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*a^4) + 1/8*(35*sqrt(b*d)*b^3*c^3*abs(b) - 45*
sqrt(b*d)*a*b^2*c^2*d*abs(b) + 9*sqrt(b*d)*a^2*b*c*d^2*abs(b) + sqrt(b*d)*a^3*d^3*abs(b))*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^4*b*c) - 1/12*(57*sqrt(b*d)*b^13*c^8*abs(b) - 394*sqrt(b*d)*a*b^12*c^7*d*abs(b) + 1170*sqrt(b*d)*a^2*b^11*
c^6*d^2*abs(b) - 1938*sqrt(b*d)*a^3*b^10*c^5*d^3*abs(b) + 1940*sqrt(b*d)*a^4*b^9*c^4*d^4*abs(b) - 1182*sqrt(b*
d)*a^5*b^8*c^3*d^5*abs(b) + 414*sqrt(b*d)*a^6*b^7*c^2*d^6*abs(b) - 70*sqrt(b*d)*a^7*b^6*c*d^7*abs(b) + 3*sqrt(
b*d)*a^8*b^5*d^8*abs(b) - 285*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11
*c^7*abs(b) + 1071*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^10*c^6*d*ab
s(b) - 1161*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^9*c^5*d^2*abs(b)
- 381*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^8*c^4*d^3*abs(b) + 16
89*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^7*c^3*d^4*abs(b) - 1251*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^6*c^2*d^5*abs(b) + 333*sqrt(b
*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^5*c*d^6*abs(b) - 15*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^4*d^7*abs(b) + 570*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^9*c^6*abs(b) - 888*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^8*c^5*d*abs(b) + 138*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^7*c^4*d^2*abs(b) - 432*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^6*c^3*d^3*abs(b) + 1182*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^4*b^5*c^2*d^4*abs(b) - 600*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^4*a^5*b^4*c*d^5*abs(b) + 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*a^6*b^3*d^6*abs(b) - 570*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*
b^7*c^5*abs(b) + 178*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^6*c^4*d*a
bs(b) - 60*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^5*c^3*d^2*abs(b)
- 276*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^3*b^4*c^2*d^3*abs(b) + 502
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^4*b^3*c*d^4*abs(b) - 30*sqrt(b*
d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5*b^2*d^5*abs(b) + 285*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^5*c^4*abs(b) - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^4*c^3*d*abs(b) - 108*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^2*b^3*c^2*d^2*abs(b) - 186*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^8*a^3*b^2*c*d^3*abs(b) + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^8*a^4*b*d^4*abs(b) - 57*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^10*b^3*c^3*abs(b) + 39*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a
*b^2*c^2*d*abs(b) + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b*c*d^
2*abs(b) - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^3*d^3*abs(b))/((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)^3*a^4*c)
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maple [B] time = 0.03, size = 707, normalized size = 2.93 \[ \frac {\sqrt {d x +c}\, \left (3 a^{3} b \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+27 a^{2} b^{2} c \,d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-135 a \,b^{3} c^{2} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 b^{4} c^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 a^{4} d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+27 a^{3} b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-135 a^{2} b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 a \,b^{3} c^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,d^{2} x^{3}+200 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c d \,x^{3}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{2} x^{3}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{2} x^{2}+76 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c d \,x^{2}-70 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c d x +28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,c^{2} x -16 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {b x +a}\, a^{4} c \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(3/2)/x^4/(b*x+a)^(3/2),x)
[Out]
1/48*(d*x+c)^(1/2)*(3*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*d^3+27*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^2*c*d^2-135*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^3*c^2*d+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*b^4*c^3+3*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*d^3+27*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*c*d^2-135*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^2*c^2*d+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^3*a*b^3*c^3-6*x^3*a^2*b*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+200*x^3*a*b^2*c*d*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-210*x^3*b^3*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x^2*a^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+76*x^2*a^2*b*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-70*x^2*a*b^2*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)-28*x*a^3*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+28*x*a^2*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-16*a^3*c
^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c/a^4/((b*x+a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)/(b*x+a)^(1/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(3/2)/x^4/(b*x+a)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)),x)
[Out]
int((c + d*x)^(3/2)/(x^4*(a + b*x)^(3/2)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**(3/2)/x**4/(b*x+a)**(3/2),x)
[Out]
Timed out
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