3.8.50 \(\int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=318 \[ -\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {c \sqrt {c+d x} (9 b c-11 a d)}{24 a^2 x^3 \sqrt {a+b x}}-\frac {5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{11/2} c^{3/2}}+\frac {\sqrt {c+d x} \left (15 a^2 d^2-322 a b c d+315 b^2 c^2\right ) (b c-a d)}{192 a^4 c x \sqrt {a+b x}}+\frac {b \sqrt {c+d x} \left (-15 a^3 d^3+839 a^2 b c d^2-1785 a b^2 c^2 d+945 b^3 c^3\right )}{192 a^5 c \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}} \]

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Rubi [A]  time = 0.36, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {98, 149, 151, 152, 12, 93, 208} \begin {gather*} \frac {\sqrt {c+d x} \left (15 a^2 d^2-322 a b c d+315 b^2 c^2\right ) (b c-a d)}{192 a^4 c x \sqrt {a+b x}}+\frac {b \sqrt {c+d x} \left (839 a^2 b c d^2-15 a^3 d^3-1785 a b^2 c^2 d+945 b^3 c^3\right )}{192 a^5 c \sqrt {a+b x}}-\frac {5 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{11/2} c^{3/2}}-\frac {\sqrt {c+d x} (63 b c-59 a d) (b c-a d)}{96 a^3 x^2 \sqrt {a+b x}}+\frac {c \sqrt {c+d x} (9 b c-11 a d)}{24 a^2 x^3 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(945*b^3*c^3 - 1785*a*b^2*c^2*d + 839*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c + d*x])/(192*a^5*c*Sqrt[a + b*x]) +
(c*(9*b*c - 11*a*d)*Sqrt[c + d*x])/(24*a^2*x^3*Sqrt[a + b*x]) - ((63*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/
(96*a^3*x^2*Sqrt[a + b*x]) + ((b*c - a*d)*(315*b^2*c^2 - 322*a*b*c*d + 15*a^2*d^2)*Sqrt[c + d*x])/(192*a^4*c*x
*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*Sqrt[a + b*x]) - (5*(b*c - a*d)^2*(63*b^2*c^2 - 14*a*b*c*d - a^
2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(11/2)*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^{5/2}}{x^5 (a+b x)^{3/2}} \, dx &=-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (9 b c-11 a d)+d (3 b c-4 a d) x\right )}{x^4 (a+b x)^{3/2}} \, dx}{4 a}\\ &=\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {\int \frac {-\frac {1}{4} c (63 b c-59 a d) (b c-a d)-\frac {3}{2} d (9 b c-8 a d) (b c-a d) x}{x^3 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{12 a^2}\\ &=\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}+\frac {\int \frac {-\frac {1}{8} c (b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right )-\frac {1}{2} b c d (63 b c-59 a d) (b c-a d) x}{x^2 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{24 a^3 c}\\ &=\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {\int \frac {-\frac {15}{16} c (b c-a d)^2 \left (63 b^2 c^2-14 a b c d-a^2 d^2\right )-\frac {1}{8} b c d (b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) x}{x (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{24 a^4 c^2}\\ &=\frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {\int -\frac {15 c (b c-a d)^3 \left (63 b^2 c^2-14 a b c d-a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a^5 c^2 (b c-a d)}\\ &=\frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}+\frac {\left (5 (b c-a d)^2 \left (63 b^2 c^2-14 a b c d-a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^5 c}\\ &=\frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}+\frac {\left (5 (b c-a d)^2 \left (63 b^2 c^2-14 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 )}{64 a^5 c}\\ &=\frac {b \left (945 b^3 c^3-1785 a b^2 c^2 d+839 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x}}{192 a^5 c \sqrt {a+b x}}+\frac {c (9 b c-11 a d) \sqrt {c+d x}}{24 a^2 x^3 \sqrt {a+b x}}-\frac {(63 b c-59 a d) (b c-a d) \sqrt {c+d x}}{96 a^3 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (315 b^2 c^2-322 a b c d+15 a^2 d^2\right ) \sqrt {c+d x}}{192 a^4 c x \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 \sqrt {a+b x}}-\frac {5 (b c-a d)^2 \left (63 b^2 c^2-14 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 )}{64 a^{11/2} c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 208, normalized size = 0.65 \begin {gather*} \frac {8 a^{7/2} x (c+d x)^{7/2} (a d+9 b c)-48 a^{9/2} c (c+d x)^{7/2}-x^2 \left (-a^2 d^2-14 a b c d+63 b^2 c^2\right ) \left (2 a^{5/2} (c+d x)^{5/2}-5 x (b c-a d) \left (\sqrt {a} \sqrt {c+d x} (a (c-2 d x)+3 b c x)-3 \sqrt {c} x \sqrt {a+b x} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{192 a^{11/2} c^2 x^4 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*a^(9/2)*c*(c + d*x)^(7/2) + 8*a^(7/2)*(9*b*c + a*d)*x*(c + d*x)^(7/2) - (63*b^2*c^2 - 14*a*b*c*d - a^2*d^
2)*x^2*(2*a^(5/2)*(c + d*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*(3*b*c*x + a*(c - 2*d*x)) - 3*Sqrt[
c]*(b*c - a*d)*x*Sqrt[a + b*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(192*a^(11/2)*c^2*x
^4*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.76, size = 393, normalized size = 1.24 \begin {gather*} \frac {5 (a d-b c)^2 \left (a^2 d^2+14 a b c d-63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{11/2} c^{3/2}}-\frac {\sqrt {c+d x} (a d-b c)^2 \left (\frac {15 a^5 d^2 (c+d x)^3}{(a+b x)^3}-\frac {384 a^4 b^2 c (c+d x)^4}{(a+b x)^4}+\frac {73 a^4 c d^2 (c+d x)^2}{(a+b x)^2}-\frac {558 a^4 b c d (c+d x)^3}{(a+b x)^3}+\frac {2511 a^3 b^2 c^2 (c+d x)^3}{(a+b x)^3}-\frac {55 a^3 c^2 d^2 (c+d x)}{a+b x}+\frac {1022 a^3 b c^2 d (c+d x)^2}{(a+b x)^2}-\frac {4599 a^2 b^2 c^3 (c+d x)^2}{(a+b x)^2}-\frac {770 a^2 b c^3 d (c+d x)}{a+b x}+15 a^2 c^3 d^2+\frac {3465 a b^2 c^4 (c+d x)}{a+b x}+210 a b c^4 d-945 b^2 c^5\right )}{192 a^5 c \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*((-(b*c) + a*d)^2*Sqrt[c + d*x]*(-945*b^2*c^5 + 210*a*b*c^4*d + 15*a^2*c^3*d^2 + (3465*a*b^2*c^4*(c + d
*x))/(a + b*x) - (770*a^2*b*c^3*d*(c + d*x))/(a + b*x) - (55*a^3*c^2*d^2*(c + d*x))/(a + b*x) - (4599*a^2*b^2*
c^3*(c + d*x)^2)/(a + b*x)^2 + (1022*a^3*b*c^2*d*(c + d*x)^2)/(a + b*x)^2 + (73*a^4*c*d^2*(c + d*x)^2)/(a + b*
x)^2 + (2511*a^3*b^2*c^2*(c + d*x)^3)/(a + b*x)^3 - (558*a^4*b*c*d*(c + d*x)^3)/(a + b*x)^3 + (15*a^5*d^2*(c +
d*x)^3)/(a + b*x)^3 - (384*a^4*b^2*c*(c + d*x)^4)/(a + b*x)^4))/(a^5*c*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a +
b*x))^4) + (5*(-(b*c) + a*d)^2*(-63*b^2*c^2 + 14*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*
Sqrt[a + b*x])])/(64*a^(11/2)*c^(3/2))

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fricas [A]  time = 13.62, size = 836, normalized size = 2.63 \begin {gather*} \left [-\frac {15 \, {\left ({\left (63 \, b^{5} c^{4} - 140 \, a b^{4} c^{3} d + 90 \, a^{2} b^{3} c^{2} d^{2} - 12 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{5} + {\left (63 \, a b^{4} c^{4} - 140 \, a^{2} b^{3} c^{3} d + 90 \, a^{3} b^{2} c^{2} d^{2} - 12 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{4}\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 (48 \, a^{5} c^{4} - {\left (945 \, a b^{4} c^{4} - 1785 \, a^{2} b^{3} c^{3} d + 839 \, a^{3} b^{2} c^{2} d^{2} - 15 \, a^{4} b c d^{3}\right )} x^{4} - {\left (315 \, a^{2} b^{3} c^{4} - 637 \, a^{3} b^{2} c^{3} d + 337 \, a^{4} b c^{2} d^{2} - 15 \, a^{5} c d^{3}\right )} x^{3} + 2 \, {\left (63 \, a^{3} b^{2} c^{4} - 122 \, a^{4} b c^{3} d + 59 \, a^{5} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (9 \, a^{4} b c^{4} - 17 \, a^{5} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{6} b c^{2} x^{5} + a^{7} c^{2} x^{4}\right )}}, \frac {15 \, {\left ({\left (63 \, b^{5} c^{4} - 140 \, a b^{4} c^{3} d + 90 \, a^{2} b^{3} c^{2} d^{2} - 12 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{5} + {\left (63 \, a b^{4} c^{4} - 140 \, a^{2} b^{3} c^{3} d + 90 \, a^{3} b^{2} c^{2} d^{2} - 12 \, a^{4} b c d^{3} - a^{5} d^{4}\right )} x^{4}\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 (48 \, a^{5} c^{4} - {\left (945 \, a b^{4} c^{4} - 1785 \, a^{2} b^{3} c^{3} d + 839 \, a^{3} b^{2} c^{2} d^{2} - 15 \, a^{4} b c d^{3}\right )} x^{4} - {\left (315 \, a^{2} b^{3} c^{4} - 637 \, a^{3} b^{2} c^{3} d + 337 \, a^{4} b c^{2} d^{2} - 15 \, a^{5} c d^{3}\right )} x^{3} + 2 \, {\left (63 \, a^{3} b^{2} c^{4} - 122 \, a^{4} b c^{3} d + 59 \, a^{5} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (9 \, a^{4} b c^{4} - 17 \, a^{5} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{6} b c^{2} x^{5} + a^{7} c^{2} x^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4*b*d^4)*x^5 + (63*a*b^
4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*x^4)*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*(48*a^5*c^4 - (945*a*b^4*c^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*
d^3)*x^4 - (315*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c*d^3)*x^3 + 2*(63*a^3*b^2*c^4 -
122*a^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8*(9*a^4*b*c^4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*
c^2*x^5 + a^7*c^2*x^4), 1/384*(15*((63*b^5*c^4 - 140*a*b^4*c^3*d + 90*a^2*b^3*c^2*d^2 - 12*a^3*b^2*c*d^3 - a^4
*b*d^4)*x^5 + (63*a*b^4*c^4 - 140*a^2*b^3*c^3*d + 90*a^3*b^2*c^2*d^2 - 12*a^4*b*c*d^3 - a^5*d^4)*x^4)*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*(48*a^5*c^4 - (945*a*b^4*c^4 - 1785*a^2*b^3*c^3*d + 839*a^3*b^2*c^2*d^2 - 15*a^4*b*c*d^3)*
x^4 - (315*a^2*b^3*c^4 - 637*a^3*b^2*c^3*d + 337*a^4*b*c^2*d^2 - 15*a^5*c*d^3)*x^3 + 2*(63*a^3*b^2*c^4 - 122*a
^4*b*c^3*d + 59*a^5*c^2*d^2)*x^2 - 8*(9*a^4*b*c^4 - 17*a^5*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b*c^2*x
^5 + a^7*c^2*x^4)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.03, size = 982, normalized size = 3.09 \begin {gather*} \frac {\sqrt {d x +c}\, \left (15 a^{4} b \,d^{4} x^{5} \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 )+180 a^{3} b^{2} c \,d^{3} x^{5} \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 )-1350 a^{2} b^{3} c^{2} d^{2} x^{5} \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 )+2100 a \,b^{4} c^{3} d \,x^{5} \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 )-945 b^{5} c^{4} x^{5} \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 )+15 a^{5} d^{4} 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 )+180 a^{4} b c \,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 )-1350 a^{3} b^{2} c^{2} 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 )+2100 a^{2} b^{3} c^{3} 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 )-945 a \,b^{4} c^{4} 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 )-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b \,d^{3} x^{4}+1678 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c \,d^{2} x^{4}-3570 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{3} c^{2} d \,x^{4}+1890 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} c^{3} x^{4}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} d^{3} x^{3}+674 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b c \,d^{2} x^{3}-1274 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c^{2} d \,x^{3}+630 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{3} c^{3} x^{3}-236 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c \,d^{2} x^{2}+488 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b \,c^{2} d \,x^{2}-252 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c^{3} x^{2}-272 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{2} d x +144 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b \,c^{3} x -96 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {b x +a}\, a^{5} c \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(d*x+c)^(1/2)*(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^5*a^4*b*d^4+180*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^2*c*d^3-1350*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^3*c^2*d^2+2100*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^4*c^3*d-945*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^5*c^
4+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*d^4+180*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*c*d^3-1350*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^2*c^2*d^2+2100*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^3*c^3*d-945*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^4*c^4-30*x^4*a^3*b*d^3*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1678*x^4*a^2*b^2*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-3570*x^4*a*b^3*
c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1890*x^4*b^4*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^3*a^4*d^3*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+674*x^3*a^3*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-1274*x^3*a^2*b^2*c
^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+630*x^3*a*b^3*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-236*x^2*a^4*c*d
^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+488*x^2*a^3*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-252*x^2*a^2*b^2
*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-272*x*a^4*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+144*x*a^3*b*c^3*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-96*a^4*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c/a^5/((b*x+a)*(d*x+c))^(1/
2)/x^4/(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^5/(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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^5\,{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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