3.7.65 \(\int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx\) [665]

Optimal. Leaf size=348 \[ -\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {5 (b c-a d)^6 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{9/2}} \]

[Out]

-1/24*(-a*d+b*c)*(a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^3/d-1/12*(a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b^2/d+
1/7*(b*x+a)^(7/2)*(d*x+c)^(7/2)/b/d+5/1024*(-a*d+b*c)^6*(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c
)^(1/2))/b^(9/2)/d^(9/2)+5/1536*(-a*d+b*c)^4*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^4/d^3-1/384*(-a*d+b*c)^3*
(a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^4/d^2-1/64*(-a*d+b*c)^2*(a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^4/d-5/
1024*(-a*d+b*c)^5*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^4

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Rubi [A]
time = 0.17, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \begin {gather*} \frac {5 (a d+b c) (b c-a d)^6 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c) (b c-a d)^5}{1024 b^4 d^4}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (a d+b c) (b c-a d)^4}{1536 b^4 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (a d+b c) (b c-a d)^3}{384 b^4 d^2}-\frac {(a+b x)^{7/2} \sqrt {c+d x} (a d+b c) (b c-a d)^2}{64 b^4 d}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (a d+b c) (b c-a d)}{24 b^3 d}-\frac {(a+b x)^{7/2} (c+d x)^{5/2} (a d+b c)}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)^5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^4) + (5*(b*c - a*d)^4*(b*c + a*d)*(a +
b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^3) - ((b*c - a*d)^3*(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^4*
d^2) - ((b*c - a*d)^2*(b*c + a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(b*c + a*d)*(a + b*
x)^(7/2)*(c + d*x)^(3/2))/(24*b^3*d) - ((b*c + a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(12*b^2*d) + ((a + b*x)^(
7/2)*(c + d*x)^(7/2))/(7*b*d) + (5*(b*c - a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(1024*b^(9/2)*d^(9/2))

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 81

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

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 x (a+b x)^{5/2} (c+d x)^{5/2} \, dx &=\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(b c+a d) \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx}{2 b d}\\ &=-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {\left (5 \left (c^2-\frac {a^2 d^2}{b^2}\right )\right ) \int (a+b x)^{5/2} (c+d x)^{3/2} \, dx}{24 d}\\ &=-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {\left ((b c-a d)^2 (b c+a d)\right ) \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{16 b^3 d}\\ &=-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {\left ((b c-a d)^3 (b c+a d)\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{128 b^4 d}\\ &=-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {\left (5 (b c-a d)^4 (b c+a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{768 b^4 d^2}\\ &=\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {\left (5 (b c-a d)^5 (b c+a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{1024 b^4 d^3}\\ &=-\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {\left (5 (b c-a d)^6 (b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2048 b^4 d^4}\\ &=-\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {\left (5 (b c-a d)^6 (b c+a d)\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 )}{1024 b^5 d^4}\\ &=-\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {\left (5 (b c-a d)^6 (b c+a d)\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 )}{1024 b^5 d^4}\\ &=-\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {5 (b c-a d)^6 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.82, size = 363, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^6 d^6+70 a^5 b d^5 (7 c+d x)-7 a^4 b^2 d^4 \left (113 c^2+46 c d x+8 d^2 x^2\right )+4 a^3 b^3 d^3 \left (75 c^3+127 c^2 d x+64 c d^2 x^2+12 d^3 x^3\right )+a^2 b^4 d^2 \left (-791 c^4+508 c^3 d x+9840 c^2 d^2 x^2+12752 c d^3 x^3+4736 d^4 x^4\right )+2 a b^5 d \left (245 c^5-161 c^4 d x+128 c^3 d^2 x^2+6376 c^2 d^3 x^3+9344 c d^4 x^4+3712 d^5 x^5\right )+b^6 \left (-105 c^6+70 c^5 d x-56 c^4 d^2 x^2+48 c^3 d^3 x^3+4736 c^2 d^4 x^4+7424 c d^5 x^5+3072 d^6 x^6\right )\right )}{21504 b^4 d^4}+\frac {5 (b c-a d)^6 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{9/2} d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^6*d^6 + 70*a^5*b*d^5*(7*c + d*x) - 7*a^4*b^2*d^4*(113*c^2 + 46*c*d*x + 8*
d^2*x^2) + 4*a^3*b^3*d^3*(75*c^3 + 127*c^2*d*x + 64*c*d^2*x^2 + 12*d^3*x^3) + a^2*b^4*d^2*(-791*c^4 + 508*c^3*
d*x + 9840*c^2*d^2*x^2 + 12752*c*d^3*x^3 + 4736*d^4*x^4) + 2*a*b^5*d*(245*c^5 - 161*c^4*d*x + 128*c^3*d^2*x^2
+ 6376*c^2*d^3*x^3 + 9344*c*d^4*x^4 + 3712*d^5*x^5) + b^6*(-105*c^6 + 70*c^5*d*x - 56*c^4*d^2*x^2 + 48*c^3*d^3
*x^3 + 4736*c^2*d^4*x^4 + 7424*c*d^5*x^5 + 3072*d^6*x^6)))/(21504*b^4*d^4) + (5*(b*c - a*d)^6*(b*c + a*d)*ArcT
anh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(1024*b^(9/2)*d^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1320\) vs. \(2(292)=584\).
time = 0.08, size = 1321, normalized size = 3.80

method	result	size
default	\(\text {Expression too large to display}\)	\(1321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43008*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-112*b^6*c^4*d^2*x^2*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+980*a*b^5*c^5*d*
(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+14848*a*b^5*d^6*x^5*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+14848*b^6*c*d^5*x^
5*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+9472*a^2*b^4*d^6*x^4*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+9472*b^6*c^2*d^
4*x^4*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+96*a^3*b^3*d^6*x^3*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+96*b^6*c^3*d^
3*x^3*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-112*a^4*b^2*d^6*x^2*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+600*(b*d)^(1
/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^3*c^3*d^3+140*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^5*b*d^6*x+140*(b*d)^(1/2
)*((d*x+c)*(b*x+a))^(1/2)*b^6*c^5*d*x-1582*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^4*c^4*d^2+980*(b*d)^(1/2)
*((d*x+c)*(b*x+a))^(1/2)*a^5*b*c*d^5-1582*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b^2*c^2*d^4-210*(b*d)^(1/2)*
((d*x+c)*(b*x+a))^(1/2)*a^6*d^6-210*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^6*c^6+105*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^7*d^7+105*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^7*c^7+1016*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^4*c^3*d^3*x-644*(b*d)^(1/2)
*((d*x+c)*(b*x+a))^(1/2)*a*b^5*c^4*d^2*x-644*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*b^2*c*d^5*x+1016*(b*d)^(1
/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b^3*c^2*d^4*x-525*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^6*b*c*d^6+945*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^
5*b^2*c^2*d^5-525*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*b^3*c^3*d^4-
525*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^4*c^4*d^3+945*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^5*c^5*d^2-525*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^6*c^6*d+6144*b^6*d^6*x^6*(b*d)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)+37376*a*b^5*c*d^5*x^4*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+25504*a^2*b^4*c*d^5*x^3*(b*d)^(1/2)*((d*x+c)*
(b*x+a))^(1/2)+25504*a*b^5*c^2*d^4*x^3*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+512*a^3*b^3*c*d^5*x^2*(b*d)^(1/2)*(
(d*x+c)*(b*x+a))^(1/2)+19680*a^2*b^4*c^2*d^4*x^2*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+512*a*b^5*c^3*d^3*x^2*(b*
d)^(1/2)*((d*x+c)*(b*x+a))^(1/2))/b^4/d^4/((d*x+c)*(b*x+a))^(1/2)/(b*d)^(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(x*(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 = 1.63, size = 1102, normalized size = 3.17 \begin {gather*} \left [\frac {105 \, {\left (b^{7} c^{7} - 5 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} - 5 \, a^{6} b c d^{6} + a^{7} d^{7}\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 (3072 \, b^{7} d^{7} x^{6} - 105 \, b^{7} c^{6} d + 490 \, a b^{6} c^{5} d^{2} - 791 \, a^{2} b^{5} c^{4} d^{3} + 300 \, a^{3} b^{4} c^{3} d^{4} - 791 \, a^{4} b^{3} c^{2} d^{5} + 490 \, a^{5} b^{2} c d^{6} - 105 \, a^{6} b d^{7} + 7424 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (3 \, b^{7} c^{3} d^{4} + 797 \, a b^{6} c^{2} d^{5} + 797 \, a^{2} b^{5} c d^{6} + 3 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (7 \, b^{7} c^{4} d^{3} - 32 \, a b^{6} c^{3} d^{4} - 1230 \, a^{2} b^{5} c^{2} d^{5} - 32 \, a^{3} b^{4} c d^{6} + 7 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (35 \, b^{7} c^{5} d^{2} - 161 \, a b^{6} c^{4} d^{3} + 254 \, a^{2} b^{5} c^{3} d^{4} + 254 \, a^{3} b^{4} c^{2} d^{5} - 161 \, a^{4} b^{3} c d^{6} + 35 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{86016 \, b^{5} d^{5}}, -\frac {105 \, {\left (b^{7} c^{7} - 5 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} - 5 \, a^{6} b c d^{6} + a^{7} d^{7}\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 (3072 \, b^{7} d^{7} x^{6} - 105 \, b^{7} c^{6} d + 490 \, a b^{6} c^{5} d^{2} - 791 \, a^{2} b^{5} c^{4} d^{3} + 300 \, a^{3} b^{4} c^{3} d^{4} - 791 \, a^{4} b^{3} c^{2} d^{5} + 490 \, a^{5} b^{2} c d^{6} - 105 \, a^{6} b d^{7} + 7424 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (3 \, b^{7} c^{3} d^{4} + 797 \, a b^{6} c^{2} d^{5} + 797 \, a^{2} b^{5} c d^{6} + 3 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (7 \, b^{7} c^{4} d^{3} - 32 \, a b^{6} c^{3} d^{4} - 1230 \, a^{2} b^{5} c^{2} d^{5} - 32 \, a^{3} b^{4} c d^{6} + 7 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (35 \, b^{7} c^{5} d^{2} - 161 \, a b^{6} c^{4} d^{3} + 254 \, a^{2} b^{5} c^{3} d^{4} + 254 \, a^{3} b^{4} c^{2} d^{5} - 161 \, a^{4} b^{3} c d^{6} + 35 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{43008 \, b^{5} d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/86016*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2
*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*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*(3072*b^7*d^7*x^6 - 105*b^7*c
^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d^6
- 105*a^6*b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 146*a*b^6*c*d^6 + 37*a^2*b^5*d^7)*
x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^5 + 797*a^2*b^5*c*d^6 + 3*a^3*b^4*d^7)*x^3 - 8*(7*b^7*c^4*d^3 - 32*a
*b^6*c^3*d^4 - 1230*a^2*b^5*c^2*d^5 - 32*a^3*b^4*c*d^6 + 7*a^4*b^3*d^7)*x^2 + 2*(35*b^7*c^5*d^2 - 161*a*b^6*c^
4*d^3 + 254*a^2*b^5*c^3*d^4 + 254*a^3*b^4*c^2*d^5 - 161*a^4*b^3*c*d^6 + 35*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(b^5*d^5), -1/43008*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^
3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*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*(3072*b^7*d^7*x^6 - 105*b^7
*c^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d
^6 - 105*a^6*b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 146*a*b^6*c*d^6 + 37*a^2*b^5*d^7
)*x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^5 + 797*a^2*b^5*c*d^6 + 3*a^3*b^4*d^7)*x^3 - 8*(7*b^7*c^4*d^3 - 32
*a*b^6*c^3*d^4 - 1230*a^2*b^5*c^2*d^5 - 32*a^3*b^4*c*d^6 + 7*a^4*b^3*d^7)*x^2 + 2*(35*b^7*c^5*d^2 - 161*a*b^6*
c^4*d^3 + 254*a^2*b^5*c^3*d^4 + 254*a^3*b^4*c^2*d^5 - 161*a^4*b^3*c*d^6 + 35*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqr
t(d*x + c))/(b^5*d^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4150 vs. \(2 (292) = 584\).
time = 2.08, size = 4150, normalized size = 11.93 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*(1680*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 -
25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^
3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a
*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*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^3))*a*c^2*abs(b) + 13440*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b
*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3
- 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*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*d^2))*a^2*c^2*abs(b)/b + 56*(sqrt(b^2*c + (b*x +
a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7
*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a
^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21
*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6
*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*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^3*d^4))*b*c^2*abs(b) + 336*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(
2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26
*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 44
7*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*
b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 +
10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*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^3*d^4))*a*c*d*abs(b) + 8960*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(
b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^
5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + s
qrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a^3*c*d*abs(b)/b^2 + 3360*(sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d
^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^
5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c
*d^3 - 35*a^4*d^4)*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^3
))*a^2*c*d*abs(b)/b + 28*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^
5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^3
4*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x +
a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)
/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8
+ 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^
2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(abs(-sqrt(b*d)*sq
rt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*b*c*d*abs(b) + 42*(sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10
)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*
d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7
+ 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^
5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b
^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*
b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*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^4*d^5))*a*d^2*abs(b) + 560*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)
*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^1
1*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*s
qrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6...

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

[Out]

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

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