\(\int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\) [784]

Optimal result

Integrand size = 22, antiderivative size = 351 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}} \]

[Out]

5/4*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(9/2)+2*a*x
^4/b/(-a*d+b*c)/(d*x+c)^(3/2)/(b*x+a)^(1/2)-2/3*c*(3*a*d+b*c)*x^3*(b*x+a)^(1/2)/b/d/(-a*d+b*c)^2/(d*x+c)^(3/2)
-2/3*c*(-3*a^2*d^2-12*a*b*c*d+7*b^2*c^2)*x^2*(b*x+a)^(1/2)/b/d^2/(-a*d+b*c)^3/(d*x+c)^(1/2)-1/12*(105*b^4*c^4-
190*a*b^3*c^3*d+36*a^2*b^2*c^2*d^2+30*a^3*b*c*d^3-45*a^4*d^4-2*b*d*(-15*a^3*d^3+9*a^2*b*c*d^2-61*a*b^2*c^2*d+3
5*b^3*c^3)*x)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^4/(-a*d+b*c)^3

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 152, 65, 223, 212} \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{3 b d^2 \sqrt {c+d x} (b c-a d)^3}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \]

[In]

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

[Out]

(2*a*x^4)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b*c + 3*a*d)*x^3*Sqrt[a + b*x])/(3*b*d*(b*c -
a*d)^2*(c + d*x)^(3/2)) - (2*c*(7*b^2*c^2 - 12*a*b*c*d - 3*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b*d^2*(b*c - a*d)^3*
Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 30*a^3*b*c
*d^3 - 45*a^4*d^4 - 2*b*d*(35*b^3*c^3 - 61*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*x))/(12*b^3*d^4*(b*c - a*
d)^3) + (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^
(7/2)*d^(9/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 155

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \int \frac {x^3 \left (4 a c+\frac {1}{2} (-b c+5 a d) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b (b c-a d)} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 \int \frac {x^2 \left (\frac {3}{2} a c (b c+3 a d)+\frac {1}{4} \left (7 b^2 c^2-6 a b c d+15 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 b d (b c-a d)^2} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {8 \int \frac {x \left (-\frac {1}{2} a c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right )+\frac {1}{8} \left (-35 b^3 c^3+61 a b^2 c^2 d-9 a^2 b c d^2+15 a^3 d^3\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 b d^2 (b c-a d)^3} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^3 d^4} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^4 d^4} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^4 d^4} \\ & = \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.91 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {45 a^5 d^4 (c+d x)^2+15 a^4 b d^3 (-2 c+d x) (c+d x)^2-6 a^3 b^2 d^2 (c+d x)^2 \left (6 c^2+2 c d x+d^2 x^2\right )-b^5 c^3 x \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )+a b^4 c^2 \left (-105 c^4+50 c^3 d x+237 c^2 d^2 x^2+48 c d^3 x^3-18 d^4 x^4\right )+2 a^2 b^3 c d \left (95 c^4+111 c^3 d x-6 c^2 d^2 x^2-9 c d^3 x^3+9 d^4 x^4\right )}{12 b^3 d^4 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{7/2} d^{9/2}} \]

[In]

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

[Out]

(45*a^5*d^4*(c + d*x)^2 + 15*a^4*b*d^3*(-2*c + d*x)*(c + d*x)^2 - 6*a^3*b^2*d^2*(c + d*x)^2*(6*c^2 + 2*c*d*x +
d^2*x^2) - b^5*c^3*x*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3) + a*b^4*c^2*(-105*c^4 + 50*c^3*d*x +
237*c^2*d^2*x^2 + 48*c*d^3*x^3 - 18*d^4*x^4) + 2*a^2*b^3*c*d*(95*c^4 + 111*c^3*d*x - 6*c^2*d^2*x^2 - 9*c*d^3*x
^3 + 9*d^4*x^4))/(12*b^3*d^4*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*
d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(7/2)*d^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2227\) vs. \(2(315)=630\).

Time = 1.72 (sec) , antiderivative size = 2228, normalized size of antiderivative = 6.35

method	result	size
default	\(\text {Expression too large to display}\)	\(2228\)

[In]

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

[Out]

1/24*(36*a*b^4*c^2*d^4*x^4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+48*a^3*b^2*c*d^5*x^3*(b*d)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)+36*a^2*b^3*c^2*d^4*x^3*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-96*a*b^4*c^3*d^3*x^3*(b*d)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+132*a^3*b^2*c^2*d^4*x^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*a^2*b^3*c^3*d^3*x^2*(b*d)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)-474*a*b^4*c^4*d^2*x^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+90*a^4*b*c^2*d^4*x*(b*d)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)+168*a^3*b^2*c^3*d^3*x*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-444*a^2*b^3*c^4*d^2*x*(b
*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-100*a*b^4*c^5*d*x*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-36*a^2*b^3*c*d^5*x^4*(
b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^3*b^2*d^6*x^4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-12*b^5*c^3*d^3*x^4*(
b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*a^4*b*d^6*x^3*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+42*b^5*c^4*d^2*x^3*(b*
d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+280*b^5*c^5*d*x^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-180*a^5*c*d^5*x*(b*d)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+60*a^4*b*c^3*d^3*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+72*a^3*b^2*c^4*d^2*(b*d)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)-380*a^2*b^3*c^5*d*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+90*ln(1/2*(2*b*d*x+2*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*c*d^6*x-45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c^3*d^4+210*a*b^4*c^6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*a^5*d^6*x^2*
(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*b^6*c^5*d^2*x^3+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*d^7
*x^3-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^4*d^3+210*b^5*c^
6*x*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*a^5*c^2*d^4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-45*ln(1/2*(2*b*d*x+
2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c*d^6*x^3-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^2*d^5*x^3-90*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^3*d^4*x^3+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^4*d^3*x^3+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*a^5*b*c*d^6*x^2-120*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a
^4*b^2*c^2*d^5*x^2-150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^3
*d^4*x^2+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^4*d^3*x^2+34
5*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^5*d^2*x^2-45*ln(1/2*(2*b
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c^2*d^5*x-105*ln(1/2*(2*b*d*x+2*((b*x+a
)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^3*d^4*x-210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^4*d^3*x+360*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^5*d^2*x+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)
/(b*d)^(1/2))*a*b^5*c^6*d*x-90*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3
*b^3*c^5*d^2+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^6*d-210
*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^6*c^6*d*x^2+45*ln(1/2*(2*b*d*x+
2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*d^7*x^2-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^6*c^7*x+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*a^6*c^2*d^5-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2
))*a*b^5*c^7)/(a*d-b*c)^3/(b*d)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/d^4/b^3/(d*x+c)^(3/2)/(b*x+a)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (315) = 630\).

Time = 1.08 (sec) , antiderivative size = 1960, normalized size of antiderivative = 5.58 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/48*(15*(7*a*b^5*c^7 - 15*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 2*a^4*b^2*c^4*d^3 + 3*a^5*b*c^3*d^4 - 3*a^6*c^
2*d^5 + (7*b^6*c^5*d^2 - 15*a*b^5*c^4*d^3 + 6*a^2*b^4*c^3*d^4 + 2*a^3*b^3*c^2*d^5 + 3*a^4*b^2*c*d^6 - 3*a^5*b*
d^7)*x^3 + (14*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 3*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 + 8*a^4*b^2*c^2*d^5 - 3*a
^5*b*c*d^6 - 3*a^6*d^7)*x^2 + (7*b^6*c^7 - a*b^5*c^6*d - 24*a^2*b^4*c^5*d^2 + 14*a^3*b^3*c^4*d^3 + 7*a^4*b^2*c
^3*d^4 + 3*a^5*b*c^2*d^5 - 6*a^6*c*d^6)*x)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*
b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(105*a*b^5*c^6*d - 190
*a^2*b^4*c^5*d^2 + 36*a^3*b^3*c^4*d^3 + 30*a^4*b^2*c^3*d^4 - 45*a^5*b*c^2*d^5 - 6*(b^6*c^3*d^4 - 3*a*b^5*c^2*d
^5 + 3*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 3*(7*b^6*c^4*d^3 - 16*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 + 8*a^3*b^3*
c*d^6 - 5*a^4*b^2*d^7)*x^3 + (140*b^6*c^5*d^2 - 237*a*b^5*c^4*d^3 + 12*a^2*b^4*c^3*d^4 + 66*a^3*b^3*c^2*d^5 -
45*a^5*b*d^7)*x^2 + (105*b^6*c^6*d - 50*a*b^5*c^5*d^2 - 222*a^2*b^4*c^4*d^3 + 84*a^3*b^3*c^3*d^4 + 45*a^4*b^2*
c^2*d^5 - 90*a^5*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^7*c^5*d^5 - 3*a^2*b^6*c^4*d^6 + 3*a^3*b^5*c^3*d
^7 - a^4*b^4*c^2*d^8 + (b^8*c^3*d^7 - 3*a*b^7*c^2*d^8 + 3*a^2*b^6*c*d^9 - a^3*b^5*d^10)*x^3 + (2*b^8*c^4*d^6 -
5*a*b^7*c^3*d^7 + 3*a^2*b^6*c^2*d^8 + a^3*b^5*c*d^9 - a^4*b^4*d^10)*x^2 + (b^8*c^5*d^5 - a*b^7*c^4*d^6 - 3*a^
2*b^6*c^3*d^7 + 5*a^3*b^5*c^2*d^8 - 2*a^4*b^4*c*d^9)*x), -1/24*(15*(7*a*b^5*c^7 - 15*a^2*b^4*c^6*d + 6*a^3*b^3
*c^5*d^2 + 2*a^4*b^2*c^4*d^3 + 3*a^5*b*c^3*d^4 - 3*a^6*c^2*d^5 + (7*b^6*c^5*d^2 - 15*a*b^5*c^4*d^3 + 6*a^2*b^4
*c^3*d^4 + 2*a^3*b^3*c^2*d^5 + 3*a^4*b^2*c*d^6 - 3*a^5*b*d^7)*x^3 + (14*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 3*a^2*b
^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 + 8*a^4*b^2*c^2*d^5 - 3*a^5*b*c*d^6 - 3*a^6*d^7)*x^2 + (7*b^6*c^7 - a*b^5*c^6*
d - 24*a^2*b^4*c^5*d^2 + 14*a^3*b^3*c^4*d^3 + 7*a^4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 - 6*a^6*c*d^6)*x)*sqrt(-b*d)
*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a
*b*d^2)*x)) + 2*(105*a*b^5*c^6*d - 190*a^2*b^4*c^5*d^2 + 36*a^3*b^3*c^4*d^3 + 30*a^4*b^2*c^3*d^4 - 45*a^5*b*c^
2*d^5 - 6*(b^6*c^3*d^4 - 3*a*b^5*c^2*d^5 + 3*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 3*(7*b^6*c^4*d^3 - 16*a*b^5*c^
3*d^4 + 6*a^2*b^4*c^2*d^5 + 8*a^3*b^3*c*d^6 - 5*a^4*b^2*d^7)*x^3 + (140*b^6*c^5*d^2 - 237*a*b^5*c^4*d^3 + 12*a
^2*b^4*c^3*d^4 + 66*a^3*b^3*c^2*d^5 - 45*a^5*b*d^7)*x^2 + (105*b^6*c^6*d - 50*a*b^5*c^5*d^2 - 222*a^2*b^4*c^4*
d^3 + 84*a^3*b^3*c^3*d^4 + 45*a^4*b^2*c^2*d^5 - 90*a^5*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^7*c^5*d^5
- 3*a^2*b^6*c^4*d^6 + 3*a^3*b^5*c^3*d^7 - a^4*b^4*c^2*d^8 + (b^8*c^3*d^7 - 3*a*b^7*c^2*d^8 + 3*a^2*b^6*c*d^9
- a^3*b^5*d^10)*x^3 + (2*b^8*c^4*d^6 - 5*a*b^7*c^3*d^7 + 3*a^2*b^6*c^2*d^8 + a^3*b^5*c*d^9 - a^4*b^4*d^10)*x^2
+ (b^8*c^5*d^5 - a*b^7*c^4*d^6 - 3*a^2*b^6*c^3*d^7 + 5*a^3*b^5*c^2*d^8 - 2*a^4*b^4*c*d^9)*x)]

Sympy [F]

\[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{5}}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (315) = 630\).

Time = 0.52 (sec) , antiderivative size = 982, normalized size of antiderivative = 2.80 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {4 \, a^{5} d}{{\left (\sqrt {b d} b^{3} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c d {\left | b \right |} + \sqrt {b d} a^{2} b d^{2} {\left | b \right |}\right )} {\left (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}\right )}} + \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{14} c^{5} d^{6} - 5 \, a b^{13} c^{4} d^{7} + 10 \, a^{2} b^{12} c^{3} d^{8} - 10 \, a^{3} b^{11} c^{2} d^{9} + 5 \, a^{4} b^{10} c d^{10} - a^{5} b^{9} d^{11}\right )} {\left (b x + a\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}} - \frac {7 \, b^{15} c^{6} d^{5} - 22 \, a b^{14} c^{5} d^{6} + 5 \, a^{2} b^{13} c^{4} d^{7} + 60 \, a^{3} b^{12} c^{3} d^{8} - 95 \, a^{4} b^{11} c^{2} d^{9} + 58 \, a^{5} b^{10} c d^{10} - 13 \, a^{6} b^{9} d^{11}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} - \frac {20 \, {\left (7 \, b^{16} c^{7} d^{4} - 29 \, a b^{15} c^{6} d^{5} + 43 \, a^{2} b^{14} c^{5} d^{6} - 21 \, a^{3} b^{13} c^{4} d^{7} - 15 \, a^{4} b^{12} c^{3} d^{8} + 27 \, a^{5} b^{11} c^{2} d^{9} - 15 \, a^{6} b^{10} c d^{10} + 3 \, a^{7} b^{9} d^{11}\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{17} c^{8} d^{3} - 180 \, a b^{16} c^{7} d^{4} + 360 \, a^{2} b^{15} c^{6} d^{5} - 340 \, a^{3} b^{14} c^{5} d^{6} + 110 \, a^{4} b^{13} c^{4} d^{7} + 84 \, a^{5} b^{12} c^{3} d^{8} - 112 \, a^{6} b^{11} c^{2} d^{9} + 52 \, a^{7} b^{10} c d^{10} - 9 \, a^{8} b^{9} d^{11}\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, \sqrt {b d} b^{2} d^{4} {\left | b \right |}} \]

[In]

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

[Out]

4*a^5*d/((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
- (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)) + 1/12*((3*(b*x + a)*(2*(b^14*c^5*d^6 -
5*a*b^13*c^4*d^7 + 10*a^2*b^12*c^3*d^8 - 10*a^3*b^11*c^2*d^9 + 5*a^4*b^10*c*d^10 - a^5*b^9*d^11)*(b*x + a)/(b
^15*c^5*d^7*abs(b) - 5*a*b^14*c^4*d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^
4*b^11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b)) - (7*b^15*c^6*d^5 - 22*a*b^14*c^5*d^6 + 5*a^2*b^13*c^4*d^7 + 60*a
^3*b^12*c^3*d^8 - 95*a^4*b^11*c^2*d^9 + 58*a^5*b^10*c*d^10 - 13*a^6*b^9*d^11)/(b^15*c^5*d^7*abs(b) - 5*a*b^14*
c^4*d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^4*b^11*c*d^11*abs(b) - a^5*b^1
0*d^12*abs(b))) - 20*(7*b^16*c^7*d^4 - 29*a*b^15*c^6*d^5 + 43*a^2*b^14*c^5*d^6 - 21*a^3*b^13*c^4*d^7 - 15*a^4*
b^12*c^3*d^8 + 27*a^5*b^11*c^2*d^9 - 15*a^6*b^10*c*d^10 + 3*a^7*b^9*d^11)/(b^15*c^5*d^7*abs(b) - 5*a*b^14*c^4*
d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^4*b^11*c*d^11*abs(b) - a^5*b^10*d^
12*abs(b)))*(b*x + a) - 3*(35*b^17*c^8*d^3 - 180*a*b^16*c^7*d^4 + 360*a^2*b^15*c^6*d^5 - 340*a^3*b^14*c^5*d^6
+ 110*a^4*b^13*c^4*d^7 + 84*a^5*b^12*c^3*d^8 - 112*a^6*b^11*c^2*d^9 + 52*a^7*b^10*c*d^10 - 9*a^8*b^9*d^11)/(b^
15*c^5*d^7*abs(b) - 5*a*b^14*c^4*d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^4
*b^11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/8*(7*b^2*
c^2 + 6*a*b*c*d + 3*a^2*d^2)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)
*b^2*d^4*abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {x^5}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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