3.7.90 \(\int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx\) [690]

Optimal. Leaf size=377 \[ -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}} \]

[Out]

-2/3*x^3*(b*x+a)^(5/2)/d/(d*x+c)^(3/2)+5/64*(-a*d+b*c)*(a^3*d^3+21*a^2*b*c*d^2-189*a*b^2*c^2*d+231*b^3*c^3)*ar
ctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/d^(13/2)-2/3*(-6*a*d+11*b*c)*x^2*(b*x+a)^(5/2)/d^2/
(-a*d+b*c)/(d*x+c)^(1/2)+5/96*(a^3*d^3+21*a^2*b*c*d^2-189*a*b^2*c^2*d+231*b^3*c^3)*(b*x+a)^(3/2)*(d*x+c)^(1/2)
/b/d^5/(-a*d+b*c)-1/24*(b*x+a)^(5/2)*(231*b^2*c^2-156*a*b*c*d+5*a^2*d^2-2*b*d*(-59*a*d+99*b*c)*x)*(d*x+c)^(1/2
)/b/d^4/(-a*d+b*c)-5/64*(a^3*d^3+21*a^2*b*c*d^2-189*a*b^2*c^2*d+231*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d^6

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Rubi [A]
time = 0.26, antiderivative size = 377, 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 = {99, 155, 152, 52, 65, 223, 212} \begin {gather*} -\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (5 a^2 d^2-2 b d x (99 b c-59 a d)-156 a b c d+231 b^2 c^2\right )}{24 b d^4 (b c-a d)}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{64 b d^6}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{96 b d^5 (b c-a d)}+\frac {5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}}-\frac {2 x^2 (a+b x)^{5/2} (11 b c-6 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^3*(a + b*x)^(5/2))/(3*d*(c + d*x)^(3/2)) - (2*(11*b*c - 6*a*d)*x^2*(a + b*x)^(5/2))/(3*d^2*(b*c - a*d)*S
qrt[c + d*x]) - (5*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64
*b*d^6) + (5*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d
^5*(b*c - a*d)) - ((a + b*x)^(5/2)*Sqrt[c + d*x]*(231*b^2*c^2 - 156*a*b*c*d + 5*a^2*d^2 - 2*b*d*(99*b*c - 59*a
*d)*x))/(24*b*d^4*(b*c - a*d)) + (5*(b*c - a*d)*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(3/2)*d^(13/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 99

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 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*} \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}+\frac {2 \int \frac {x^2 (a+b x)^{3/2} \left (3 a+\frac {11 b x}{2}\right )}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {4 \int \frac {x (a+b x)^{3/2} \left (-a (11 b c-6 a d)-\frac {1}{4} b (99 b c-59 a d) x\right )}{\sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{48 b d^4 (b c-a d)}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}-\frac {\left (5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b d^5}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d^6}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\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 )}{64 b^2 d^6}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\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 )}{64 b^2 d^6}\\ &=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 11.81, size = 331, normalized size = 0.88 \begin {gather*} \frac {48 x^2 (a+b x)^4+\frac {16 c (a+b x)^4 \left (3 a^2 d^2 (c+2 d x)-14 a b c d (5 c+6 d x)+11 b^2 c^2 (9 c+10 d x)\right )}{d^2 (b c-a d)^2}-\frac {\left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (c+d x)^2 \left (\sqrt {d} \sqrt {b c-a d} (a+b x) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (33 a^2 d^2+2 a b d (-20 c+13 d x)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )-15 (b c-a d)^3 \sqrt {a+b x} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{d^{11/2} (b c-a d)^{5/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}}{192 b d \sqrt {a+b x} (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(48*x^2*(a + b*x)^4 + (16*c*(a + b*x)^4*(3*a^2*d^2*(c + 2*d*x) - 14*a*b*c*d*(5*c + 6*d*x) + 11*b^2*c^2*(9*c +
10*d*x)))/(d^2*(b*c - a*d)^2) - ((231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*(c + d*x)^2*(Sqrt[
d]*Sqrt[b*c - a*d]*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*(33*a^2*d^2 + 2*a*b*d*(-20*c + 13*d*x) + b^2*(15*
c^2 - 10*c*d*x + 8*d^2*x^2)) - 15*(b*c - a*d)^3*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]
))/(d^(11/2)*(b*c - a*d)^(5/2)*Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(192*b*d*Sqrt[a + b*x]*(c + d*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1365\) vs. \(2(333)=666\).
time = 0.08, size = 1366, normalized size = 3.62

method	result	size
default	\(\text {Expression too large to display}\)	\(1366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(b*x+a)^(1/2)*(-272*a*b^2*d^5*x^4*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+176*b^3*c*d^4*x^4*(b*d)^(1/2)*((d
*x+c)*(b*x+a))^(1/2)-236*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*d^5*x^3-396*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(
1/2)*b^3*c^2*d^3*x^3-60*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*c*d^4*x+3486*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)*a^2*b*c^3*d^2-10290*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^4*d+12600*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a
))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^3*c^4*d^2*x+9240*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^3*c^4*d*
x-6300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^3*d^3*x+600*ln(1/
2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c^2*d^4*x+1386*(b*d)^(1/2)*((d*x+
c)*(b*x+a))^(1/2)*b^3*c^3*d^2*x^2+6300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*a*b^3*c^3*d^3*x^2+300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c
*d^5*x^2-3150*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^2*d^4*x^2+
15*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^6*x^2+15*ln(1/2*(2*b*d*x+
2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*c^2*d^4+6930*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)*b^3*c^5-3465*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^6+4944*(b*d)
^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c^2*d^3*x-14028*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c^3*d^2*x-3465*
ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^4*d^2*x^2-6930*ln(1/2*(2*b*d
*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^5*d*x+300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x
+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c^3*d^3-3150*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^2*c^4*d^2+6300*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*a*b^3*c^5*d-96*b^3*d^5*x^5*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-30*(b*d)^(1/2)*((d*x+c)*(b*x+a
))^(1/2)*a^3*c^2*d^3+966*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b*c*d^4*x^2-2322*(b*d)^(1/2)*((d*x+c)*(b*x+a)
)^(1/2)*a*b^2*c^2*d^3*x^2+30*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*c
*d^5*x-30*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*d^5*x^2+632*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^2*c*d^4*
x^3)/((d*x+c)*(b*x+a))^(1/2)/(b*d)^(1/2)/(d*x+c)^(3/2)/b/d^6

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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(x^3*(b*x+a)^(5/2)/(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

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Fricas [A]
time = 2.21, size = 1066, normalized size = 2.83 \begin {gather*} \left [-\frac {15 \, {\left (231 \, b^{4} c^{6} - 420 \, a b^{3} c^{5} d + 210 \, a^{2} b^{2} c^{4} d^{2} - 20 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (231 \, b^{4} c^{4} d^{2} - 420 \, a b^{3} c^{3} d^{3} + 210 \, a^{2} b^{2} c^{2} d^{4} - 20 \, a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + 2 \, {\left (231 \, b^{4} c^{5} d - 420 \, a b^{3} c^{4} d^{2} + 210 \, a^{2} b^{2} c^{3} d^{3} - 20 \, a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{4} d^{6} x^{5} - 3465 \, b^{4} c^{5} d + 5145 \, a b^{3} c^{4} d^{2} - 1743 \, a^{2} b^{2} c^{3} d^{3} + 15 \, a^{3} b c^{2} d^{4} - 8 \, {\left (11 \, b^{4} c d^{5} - 17 \, a b^{3} d^{6}\right )} x^{4} + 2 \, {\left (99 \, b^{4} c^{2} d^{4} - 158 \, a b^{3} c d^{5} + 59 \, a^{2} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (231 \, b^{4} c^{3} d^{3} - 387 \, a b^{3} c^{2} d^{4} + 161 \, a^{2} b^{2} c d^{5} - 5 \, a^{3} b d^{6}\right )} x^{2} - 6 \, {\left (770 \, b^{4} c^{4} d^{2} - 1169 \, a b^{3} c^{3} d^{3} + 412 \, a^{2} b^{2} c^{2} d^{4} - 5 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (b^{2} d^{9} x^{2} + 2 \, b^{2} c d^{8} x + b^{2} c^{2} d^{7}\right )}}, -\frac {15 \, {\left (231 \, b^{4} c^{6} - 420 \, a b^{3} c^{5} d + 210 \, a^{2} b^{2} c^{4} d^{2} - 20 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (231 \, b^{4} c^{4} d^{2} - 420 \, a b^{3} c^{3} d^{3} + 210 \, a^{2} b^{2} c^{2} d^{4} - 20 \, a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + 2 \, {\left (231 \, b^{4} c^{5} d - 420 \, a b^{3} c^{4} d^{2} + 210 \, a^{2} b^{2} c^{3} d^{3} - 20 \, a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{6} x^{5} - 3465 \, b^{4} c^{5} d + 5145 \, a b^{3} c^{4} d^{2} - 1743 \, a^{2} b^{2} c^{3} d^{3} + 15 \, a^{3} b c^{2} d^{4} - 8 \, {\left (11 \, b^{4} c d^{5} - 17 \, a b^{3} d^{6}\right )} x^{4} + 2 \, {\left (99 \, b^{4} c^{2} d^{4} - 158 \, a b^{3} c d^{5} + 59 \, a^{2} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (231 \, b^{4} c^{3} d^{3} - 387 \, a b^{3} c^{2} d^{4} + 161 \, a^{2} b^{2} c d^{5} - 5 \, a^{3} b d^{6}\right )} x^{2} - 6 \, {\left (770 \, b^{4} c^{4} d^{2} - 1169 \, a b^{3} c^{3} d^{3} + 412 \, a^{2} b^{2} c^{2} d^{4} - 5 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (b^{2} d^{9} x^{2} + 2 \, b^{2} c d^{8} x + b^{2} c^{2} d^{7}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(231*b^4*c^6 - 420*a*b^3*c^5*d + 210*a^2*b^2*c^4*d^2 - 20*a^3*b*c^3*d^3 - a^4*c^2*d^4 + (231*b^4*c
^4*d^2 - 420*a*b^3*c^3*d^3 + 210*a^2*b^2*c^2*d^4 - 20*a^3*b*c*d^5 - a^4*d^6)*x^2 + 2*(231*b^4*c^5*d - 420*a*b^
3*c^4*d^2 + 210*a^2*b^2*c^3*d^3 - 20*a^3*b*c^2*d^4 - a^4*c*d^5)*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*(48*b^4*d^6*x^5 - 3465*b^4*c^5*d + 5145*a*b^3*c^4*d^2 - 1743*a^2*b^2*c^3*d^3 + 15*a^3*b*c^2*d^4 - 8*(11*b^4*
c*d^5 - 17*a*b^3*d^6)*x^4 + 2*(99*b^4*c^2*d^4 - 158*a*b^3*c*d^5 + 59*a^2*b^2*d^6)*x^3 - 3*(231*b^4*c^3*d^3 - 3
87*a*b^3*c^2*d^4 + 161*a^2*b^2*c*d^5 - 5*a^3*b*d^6)*x^2 - 6*(770*b^4*c^4*d^2 - 1169*a*b^3*c^3*d^3 + 412*a^2*b^
2*c^2*d^4 - 5*a^3*b*c*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^9*x^2 + 2*b^2*c*d^8*x + b^2*c^2*d^7), -1/384
*(15*(231*b^4*c^6 - 420*a*b^3*c^5*d + 210*a^2*b^2*c^4*d^2 - 20*a^3*b*c^3*d^3 - a^4*c^2*d^4 + (231*b^4*c^4*d^2
- 420*a*b^3*c^3*d^3 + 210*a^2*b^2*c^2*d^4 - 20*a^3*b*c*d^5 - a^4*d^6)*x^2 + 2*(231*b^4*c^5*d - 420*a*b^3*c^4*d
^2 + 210*a^2*b^2*c^3*d^3 - 20*a^3*b*c^2*d^4 - a^4*c*d^5)*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*(48*b^4*d^6*x^5 - 3465*b
^4*c^5*d + 5145*a*b^3*c^4*d^2 - 1743*a^2*b^2*c^3*d^3 + 15*a^3*b*c^2*d^4 - 8*(11*b^4*c*d^5 - 17*a*b^3*d^6)*x^4
+ 2*(99*b^4*c^2*d^4 - 158*a*b^3*c*d^5 + 59*a^2*b^2*d^6)*x^3 - 3*(231*b^4*c^3*d^3 - 387*a*b^3*c^2*d^4 + 161*a^2
*b^2*c*d^5 - 5*a^3*b*d^6)*x^2 - 6*(770*b^4*c^4*d^2 - 1169*a*b^3*c^3*d^3 + 412*a^2*b^2*c^2*d^4 - 5*a^3*b*c*d^5)
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^9*x^2 + 2*b^2*c*d^8*x + b^2*c^2*d^7)]

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Sympy [F(-1)] Timed out
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(x**3*(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (333) = 666\).
time = 2.05, size = 686, normalized size = 1.82 \begin {gather*} \frac {{\left ({\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b^{5} c d^{10} {\left | b \right |} - a b^{4} d^{11} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} c d^{11} - a b^{5} d^{12}} - \frac {11 \, b^{6} c^{2} d^{9} {\left | b \right |} + 2 \, a b^{5} c d^{10} {\left | b \right |} - 13 \, a^{2} b^{4} d^{11} {\left | b \right |}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} + \frac {9 \, {\left (11 \, b^{7} c^{3} d^{8} {\left | b \right |} - 9 \, a b^{6} c^{2} d^{9} {\left | b \right |} + a^{2} b^{5} c d^{10} {\left | b \right |} - 3 \, a^{3} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (231 \, b^{8} c^{4} d^{7} {\left | b \right |} - 420 \, a b^{7} c^{3} d^{8} {\left | b \right |} + 210 \, a^{2} b^{6} c^{2} d^{9} {\left | b \right |} - 20 \, a^{3} b^{5} c d^{10} {\left | b \right |} - a^{4} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {20 \, {\left (231 \, b^{9} c^{5} d^{6} {\left | b \right |} - 651 \, a b^{8} c^{4} d^{7} {\left | b \right |} + 630 \, a^{2} b^{7} c^{3} d^{8} {\left | b \right |} - 230 \, a^{3} b^{6} c^{2} d^{9} {\left | b \right |} + 19 \, a^{4} b^{5} c d^{10} {\left | b \right |} + a^{5} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (231 \, b^{10} c^{6} d^{5} {\left | b \right |} - 882 \, a b^{9} c^{5} d^{6} {\left | b \right |} + 1281 \, a^{2} b^{8} c^{4} d^{7} {\left | b \right |} - 860 \, a^{3} b^{7} c^{3} d^{8} {\left | b \right |} + 249 \, a^{4} b^{6} c^{2} d^{9} {\left | b \right |} - 18 \, a^{5} b^{5} c d^{10} {\left | b \right |} - a^{6} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} \sqrt {b x + a}}{192 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (231 \, b^{4} c^{4} {\left | b \right |} - 420 \, a b^{3} c^{3} d {\left | b \right |} + 210 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 20 \, a^{3} b c d^{3} {\left | b \right |} - a^{4} d^{4} {\left | b \right |}\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 |}\right )}{64 \, \sqrt {b d} b^{2} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(((2*(4*(b*x + a)*(6*(b^5*c*d^10*abs(b) - a*b^4*d^11*abs(b))*(b*x + a)/(b^6*c*d^11 - a*b^5*d^12) - (11*b
^6*c^2*d^9*abs(b) + 2*a*b^5*c*d^10*abs(b) - 13*a^2*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12)) + 9*(11*b^7*c^3
*d^8*abs(b) - 9*a*b^6*c^2*d^9*abs(b) + a^2*b^5*c*d^10*abs(b) - 3*a^3*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12
))*(b*x + a) - 3*(231*b^8*c^4*d^7*abs(b) - 420*a*b^7*c^3*d^8*abs(b) + 210*a^2*b^6*c^2*d^9*abs(b) - 20*a^3*b^5*
c*d^10*abs(b) - a^4*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12))*(b*x + a) - 20*(231*b^9*c^5*d^6*abs(b) - 651*a
*b^8*c^4*d^7*abs(b) + 630*a^2*b^7*c^3*d^8*abs(b) - 230*a^3*b^6*c^2*d^9*abs(b) + 19*a^4*b^5*c*d^10*abs(b) + a^5
*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12))*(b*x + a) - 15*(231*b^10*c^6*d^5*abs(b) - 882*a*b^9*c^5*d^6*abs(b
) + 1281*a^2*b^8*c^4*d^7*abs(b) - 860*a^3*b^7*c^3*d^8*abs(b) + 249*a^4*b^6*c^2*d^9*abs(b) - 18*a^5*b^5*c*d^10*
abs(b) - a^6*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) -
5/64*(231*b^4*c^4*abs(b) - 420*a*b^3*c^3*d*abs(b) + 210*a^2*b^2*c^2*d^2*abs(b) - 20*a^3*b*c*d^3*abs(b) - a^4*
d^4*abs(b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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