3.6.81 \(\int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx\)

Optimal. Leaf size=340 \[ \frac {\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{40 c x^4} \]

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Rubi [A]  time = 0.32, antiderivative size = 340, 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 = {97, 149, 151, 12, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (61 a^2 b c d^2-35 a^3 d^3-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac {\sqrt {a+b x} \sqrt {c+d 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 )}{1920 a^3 c^4 x}+\frac {\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 )}{128 a^{7/2} c^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{40 c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

-((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*c*x^4) - (((3*b^2*c)/a + 12*b*d - (7*a*d^2)/c)*Sqrt[a + b*x]*
Sqrt[c + d*x])/(240*c*x^3) + ((15*b^3*c^3 - 9*a*b^2*c^2*d + 61*a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(960*a^2*c^3*x^2) - ((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]*Sqrt[c + d*x])/(1920*a^3*c^4*x) - ((a + b*x)^(3/2)*Sqrt[c + d*x])/(5*x^5) + ((b*c - a*d)^3*(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])])/(128*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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 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 {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx &=-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {1}{5} \int \frac {\sqrt {a+b x} \left (\frac {1}{2} (3 b c+a d)+2 b d x\right )}{x^5 \sqrt {c+d x}} \, dx\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {\int \frac {\frac {1}{4} \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right )+\frac {1}{2} b d (7 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 c}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\int \frac {\frac {1}{8} \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right )+\frac {1}{2} b d \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a c^2}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {\int \frac {\frac {1}{16} \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 )+\frac {1}{8} b d \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^2 c^3}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\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} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\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}{120 a^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\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} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\left ((b c-a d)^3 \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}{256 a^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\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} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}-\frac {\left ((b c-a d)^3 \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 )}{128 a^3 c^4}\\ &=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\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} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {(b c-a d)^3 \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 )}{128 a^{7/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 231, normalized size = 0.68 \begin {gather*} \frac {\frac {5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \left (\frac {3 x (b c-a d) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+a d x+b c x)\right )}{a^{3/2} c^{3/2}}-8 (a+b x)^{3/2} (c+d x)^{3/2}\right )}{24 c x^3}-\frac {16 a c (a+b x)^{5/2} (c+d x)^{3/2}}{x^5}+\frac {2 (a+b x)^{5/2} (c+d x)^{3/2} (7 a d+5 b c)}{x^4}}{80 a^2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

((-16*a*c*(a + b*x)^(5/2)*(c + d*x)^(3/2))/x^5 + (2*(5*b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4 + (5*
(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(-8*(a + b*x)^(3/2)*(c + d*x)^(3/2) + (3*(b*c - a*d)*x*(-(Sqrt[a]*Sqrt[c]*
Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + b*c*x + a*d*x)) + (b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt
[a]*Sqrt[c + d*x])]))/(a^(3/2)*c^(3/2))))/(24*c*x^3))/(80*a^2*c^2)

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IntegrateAlgebraic [A]  time = 0.57, size = 445, normalized size = 1.31 \begin {gather*} \frac {\sqrt {c+d x} (a d-b c)^3 \left (\frac {105 a^6 d^2 (c+d x)^4}{(a+b x)^4}-\frac {490 a^5 c d^2 (c+d x)^3}{(a+b x)^3}+\frac {90 a^5 b c d (c+d x)^4}{(a+b x)^4}+\frac {45 a^4 b^2 c^2 (c+d x)^4}{(a+b x)^4}+\frac {896 a^4 c^2 d^2 (c+d x)^2}{(a+b x)^2}-\frac {420 a^4 b c^2 d (c+d x)^3}{(a+b x)^3}-\frac {210 a^3 b^2 c^3 (c+d x)^3}{(a+b x)^3}-\frac {790 a^3 c^3 d^2 (c+d x)}{a+b x}+\frac {768 a^3 b c^3 d (c+d x)^2}{(a+b x)^2}-\frac {384 a^2 b^2 c^4 (c+d x)^2}{(a+b x)^2}+\frac {420 a^2 b c^4 d (c+d x)}{a+b x}-105 a^2 c^4 d^2+\frac {210 a b^2 c^5 (c+d x)}{a+b x}-90 a b c^5 d-45 b^2 c^6\right )}{1920 a^3 c^4 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^5}-\frac {(a d-b c)^3 \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 )}{128 a^{7/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

((-(b*c) + a*d)^3*Sqrt[c + d*x]*(-45*b^2*c^6 - 90*a*b*c^5*d - 105*a^2*c^4*d^2 + (210*a*b^2*c^5*(c + d*x))/(a +
b*x) + (420*a^2*b*c^4*d*(c + d*x))/(a + b*x) - (790*a^3*c^3*d^2*(c + d*x))/(a + b*x) - (384*a^2*b^2*c^4*(c +
d*x)^2)/(a + b*x)^2 + (768*a^3*b*c^3*d*(c + d*x)^2)/(a + b*x)^2 + (896*a^4*c^2*d^2*(c + d*x)^2)/(a + b*x)^2 -
(210*a^3*b^2*c^3*(c + d*x)^3)/(a + b*x)^3 - (420*a^4*b*c^2*d*(c + d*x)^3)/(a + b*x)^3 - (490*a^5*c*d^2*(c + d*
x)^3)/(a + b*x)^3 + (45*a^4*b^2*c^2*(c + d*x)^4)/(a + b*x)^4 + (90*a^5*b*c*d*(c + d*x)^4)/(a + b*x)^4 + (105*a
^6*d^2*(c + d*x)^4)/(a + b*x)^4))/(1920*a^3*c^4*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a + b*x))^5) - ((-(b*c) + a
*d)^3*(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])])/(128*a^(7/
2)*c^(9/2))

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fricas [A]  time = 18.41, size = 730, normalized size = 2.15 \begin {gather*} \left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (384 \, a^{5} c^{5} + {\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{4} c^{5} x^{5}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (384 \, a^{5} c^{5} + {\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{4} c^{5} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*
sqrt(a*c)*x^5*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*(384*a^5*c^5 + (45*a*b^4*c^5 - 30*a^2*b^3*c^4*d - 3
6*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^
3*d^2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2 + 48*(11*a^4*b*c^5 + a^5*
c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5), -1/3840*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d
^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*sqrt(-a*c)*x^5*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*(384*a^5*c^5 + (45*a*b^4*
c^5 - 30*a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a
^3*b^2*c^4*d + 61*a^4*b*c^3*d^2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2
+ 48*(11*a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5)]

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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((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 967, normalized size = 2.84 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-225 a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+90 a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+30 a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+45 a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-45 b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+380 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-72 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}-60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}+90 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-244 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}+36 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}-60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}+192 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}+48 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x +1056 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{3} c^{4} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*a^5*d^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2))/x)-225*a^4*b*c*d^4*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
)/x)+90*a^3*b^2*c^2*d^3*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+30*a^2*b^3
*c^3*d^2*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+45*a*b^4*c^4*d*x^5*ln((a*
d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-45*b^5*c^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4*x^4+380*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3*x^4-72*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^
2*c^2*d^2*x^4-60*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d*x^4+90*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*b^4*c^4*x^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*d^3*x^3-244*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^2*d^2*x^3+36*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^3*d*x^3-6
0*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^4*x^3-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^
4*c^2*d^2*x^2+192*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^3*d*x^2+48*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*a^2*b^2*c^4*x^2+96*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^3*d*x+1056*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^4*x+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^4*c^4)/(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)/x^5/(a*c)^(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((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,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 (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6,x)

[Out]

int(((a + b*x)^(3/2)*(c + d*x)^(1/2))/x^6, 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((b*x+a)**(3/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

Timed out

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