3.23.30 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\) [2230]

Optimal. Leaf size=167 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}} \]

[Out]

-2/7*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(7/2)-2/5*B*(b*x+a)^(5/2)/e^2/(e*x+d)^(5/2)-2/3*b*B*(b*x+a)
^(3/2)/e^3/(e*x+d)^(3/2)+2*b^(5/2)*B*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/e^(9/2)-2*b^2*B*(b*x
+a)^(1/2)/e^4/(e*x+d)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 65, 223, 212} \begin {gather*} -\frac {2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) - (2*B*(a + b*x)^(5/2))/(5*e^2*(d + e*x)^(5
/2)) - (2*b*B*(a + b*x)^(3/2))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2*B*Sqrt[a + b*x])/(e^4*Sqrt[d + e*x]) + (2*b^(5
/2)*B*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 79

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

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 {(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {B \int \frac {(a+b x)^{5/2}}{(d+e x)^{7/2}} \, dx}{e}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {(b B) \int \frac {(a+b x)^{3/2}}{(d+e x)^{5/2}} \, dx}{e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}+\frac {\left (b^2 B\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{3/2}} \, dx}{e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {\left (b^3 B\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {\left (2 b^2 B\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {\left (2 b^2 B\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{7 e (b d-a e) (d+e x)^{7/2}}-\frac {2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2 B \sqrt {a+b x}}{e^4 \sqrt {d+e x}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 215, normalized size = 1.29 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (-15 B d e^3 (a+b x)^3+15 A e^4 (a+b x)^3-21 b B d e^2 (a+b x)^2 (d+e x)+21 a B e^3 (a+b x)^2 (d+e x)-35 b^2 B d e (a+b x) (d+e x)^2+35 a b B e^2 (a+b x) (d+e x)^2-105 b^3 B d (d+e x)^3+105 a b^2 B e (d+e x)^3\right )}{105 e^4 (-b d+a e) (d+e x)^{7/2}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*Sqrt[a + b*x]*(-15*B*d*e^3*(a + b*x)^3 + 15*A*e^4*(a + b*x)^3 - 21*b*B*d*e^2*(a + b*x)^2*(d + e*x) + 21*a*
B*e^3*(a + b*x)^2*(d + e*x) - 35*b^2*B*d*e*(a + b*x)*(d + e*x)^2 + 35*a*b*B*e^2*(a + b*x)*(d + e*x)^2 - 105*b^
3*B*d*(d + e*x)^3 + 105*a*b^2*B*e*(d + e*x)^3))/(105*e^4*(-(b*d) + a*e)*(d + e*x)^(7/2)) + (2*b^(5/2)*B*ArcTan
h[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/e^(9/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1088\) vs. \(2(133)=266\).
time = 0.21, size = 1089, normalized size = 6.52

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (420 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{2} e^{3} x^{3}+630 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e^{2} x^{2}+420 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4} e x -105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{4} e +30 A \,b^{3} e^{4} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+42 B \,a^{3} e^{4} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+12 B \,a^{3} d \,e^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} e^{5} x^{4}+105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d \,e^{4} x^{4}+105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{5}+568 B a \,b^{2} d \,e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+92 B \,a^{2} b d \,e^{3} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+476 B a \,b^{2} d^{2} e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+30 A \,a^{3} e^{4} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-210 B \,b^{3} d^{4} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+154 B \,a^{2} b \,e^{4} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-812 B \,b^{3} d^{2} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+90 A \,a^{2} b \,e^{4} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-700 B \,b^{3} d^{3} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+28 B \,a^{2} b \,d^{2} e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+140 B a \,b^{2} d^{3} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-420 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d \,e^{4} x^{3}-630 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{3} x^{2}-420 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e^{2} x +322 B a \,b^{2} e^{4} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-352 B \,b^{3} d \,e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+90 A a \,b^{2} e^{4} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{105 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \left (a e -b d \right ) \sqrt {b e}\, \left (e x +d \right )^{\frac {7}{2}} e^{4}}\)	\(1089\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/105*(b*x+a)^(1/2)*(420*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^
2*e^3*x^3+630*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^3*e^2*x^2+42
0*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4*e*x-105*B*ln(1/2*(2*b*
e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^4*e+30*A*b^3*e^4*x^3*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+42*B*a^3*e^4*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12*B*a^3*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*e^5*x^4+105
*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d*e^4*x^4+105*B*ln(1/2*(2*b
*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^5+568*B*a*b^2*d*e^3*x^2*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+92*B*a^2*b*d*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+476*B*a*b^2*d^2*e^2*x*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+30*A*a^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-210*B*b^3*d^4*((b*x+a)*(e*x+d))^(1/2)
*(b*e)^(1/2)+154*B*a^2*b*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-812*B*b^3*d^2*e^2*x^2*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)+90*A*a^2*b*e^4*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-700*B*b^3*d^3*e*x*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)+28*B*a^2*b*d^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+140*B*a*b^2*d^3*e*((b*x+a)*(e*x+d))^(1/2)
*(b*e)^(1/2)-420*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d*e^4*x^3
-630*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^2*e^3*x^2-420*B*ln(
1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^3*e^2*x+322*B*a*b^2*e^4*x^3*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-352*B*b^3*d*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*a*b^2*e^4*x^2*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(7/2)/e^4

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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)^(5/2)*(B*x+A)/(e*x+d)^(9/2),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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (134) = 268\).
time = 13.62, size = 1012, normalized size = 6.06 \begin {gather*} \left [\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} x^{4} e^{5} + {\left (B b^{3} d x^{4} - 4 \, B a b^{2} d x^{3}\right )} e^{4} + 2 \, {\left (2 \, B b^{3} d^{2} x^{3} - 3 \, B a b^{2} d^{2} x^{2}\right )} e^{3} + 2 \, {\left (3 \, B b^{3} d^{3} x^{2} - 2 \, B a b^{2} d^{3} x\right )} e^{2} + {\left (4 \, B b^{3} d^{4} x - B a b^{2} d^{4}\right )} e\right )} \sqrt {b} e^{\left (-\frac {1}{2}\right )} \log \left (b^{2} d^{2} + 4 \, {\left (b d e + {\left (2 \, b x + a\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\left (-\frac {1}{2}\right )} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (105 \, B b^{3} d^{4} - {\left (15 \, A a^{3} + {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{3} + {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x\right )} e^{4} + 2 \, {\left (88 \, B b^{3} d x^{3} - 142 \, B a b^{2} d x^{2} - 23 \, B a^{2} b d x - 3 \, B a^{3} d\right )} e^{3} + 14 \, {\left (29 \, B b^{3} d^{2} x^{2} - 17 \, B a b^{2} d^{2} x - B a^{2} b d^{2}\right )} e^{2} + 70 \, {\left (5 \, B b^{3} d^{3} x - B a b^{2} d^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{210 \, {\left (b d^{5} e^{4} - a x^{4} e^{9} + {\left (b d x^{4} - 4 \, a d x^{3}\right )} e^{8} + 2 \, {\left (2 \, b d^{2} x^{3} - 3 \, a d^{2} x^{2}\right )} e^{7} + 2 \, {\left (3 \, b d^{3} x^{2} - 2 \, a d^{3} x\right )} e^{6} + {\left (4 \, b d^{4} x - a d^{4}\right )} e^{5}\right )}}, -\frac {105 \, {\left (B b^{3} d^{5} - B a b^{2} x^{4} e^{5} + {\left (B b^{3} d x^{4} - 4 \, B a b^{2} d x^{3}\right )} e^{4} + 2 \, {\left (2 \, B b^{3} d^{2} x^{3} - 3 \, B a b^{2} d^{2} x^{2}\right )} e^{3} + 2 \, {\left (3 \, B b^{3} d^{3} x^{2} - 2 \, B a b^{2} d^{3} x\right )} e^{2} + {\left (4 \, B b^{3} d^{4} x - B a b^{2} d^{4}\right )} e\right )} \sqrt {-b e^{\left (-1\right )}} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {-b e^{\left (-1\right )}}}{2 \, {\left (b^{2} d x + a b d + {\left (b^{2} x^{2} + a b x\right )} e\right )}}\right ) + 2 \, {\left (105 \, B b^{3} d^{4} - {\left (15 \, A a^{3} + {\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{3} + {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x\right )} e^{4} + 2 \, {\left (88 \, B b^{3} d x^{3} - 142 \, B a b^{2} d x^{2} - 23 \, B a^{2} b d x - 3 \, B a^{3} d\right )} e^{3} + 14 \, {\left (29 \, B b^{3} d^{2} x^{2} - 17 \, B a b^{2} d^{2} x - B a^{2} b d^{2}\right )} e^{2} + 70 \, {\left (5 \, B b^{3} d^{3} x - B a b^{2} d^{3}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{105 \, {\left (b d^{5} e^{4} - a x^{4} e^{9} + {\left (b d x^{4} - 4 \, a d x^{3}\right )} e^{8} + 2 \, {\left (2 \, b d^{2} x^{3} - 3 \, a d^{2} x^{2}\right )} e^{7} + 2 \, {\left (3 \, b d^{3} x^{2} - 2 \, a d^{3} x\right )} e^{6} + {\left (4 \, b d^{4} x - a d^{4}\right )} e^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(B*b^3*d^5 - B*a*b^2*x^4*e^5 + (B*b^3*d*x^4 - 4*B*a*b^2*d*x^3)*e^4 + 2*(2*B*b^3*d^2*x^3 - 3*B*a*b^
2*d^2*x^2)*e^3 + 2*(3*B*b^3*d^3*x^2 - 2*B*a*b^2*d^3*x)*e^2 + (4*B*b^3*d^4*x - B*a*b^2*d^4)*e)*sqrt(b)*e^(-1/2)
*log(b^2*d^2 + 4*(b*d*e + (2*b*x + a)*e^2)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(-1/2) + (8*b^2*x^2 + 8*a*b*x
+ a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) - 4*(105*B*b^3*d^4 - (15*A*a^3 + (161*B*a*b^2 + 15*A*b^3)*x^3 + (77*B
*a^2*b + 45*A*a*b^2)*x^2 + 3*(7*B*a^3 + 15*A*a^2*b)*x)*e^4 + 2*(88*B*b^3*d*x^3 - 142*B*a*b^2*d*x^2 - 23*B*a^2*
b*d*x - 3*B*a^3*d)*e^3 + 14*(29*B*b^3*d^2*x^2 - 17*B*a*b^2*d^2*x - B*a^2*b*d^2)*e^2 + 70*(5*B*b^3*d^3*x - B*a*
b^2*d^3)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(b*d^5*e^4 - a*x^4*e^9 + (b*d*x^4 - 4*a*d*x^3)*e^8 + 2*(2*b*d^2*x^3 -
3*a*d^2*x^2)*e^7 + 2*(3*b*d^3*x^2 - 2*a*d^3*x)*e^6 + (4*b*d^4*x - a*d^4)*e^5), -1/105*(105*(B*b^3*d^5 - B*a*b
^2*x^4*e^5 + (B*b^3*d*x^4 - 4*B*a*b^2*d*x^3)*e^4 + 2*(2*B*b^3*d^2*x^3 - 3*B*a*b^2*d^2*x^2)*e^3 + 2*(3*B*b^3*d^
3*x^2 - 2*B*a*b^2*d^3*x)*e^2 + (4*B*b^3*d^4*x - B*a*b^2*d^4)*e)*sqrt(-b*e^(-1))*arctan(1/2*(b*d + (2*b*x + a)*
e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(-b*e^(-1))/(b^2*d*x + a*b*d + (b^2*x^2 + a*b*x)*e)) + 2*(105*B*b^3*d^4 - (
15*A*a^3 + (161*B*a*b^2 + 15*A*b^3)*x^3 + (77*B*a^2*b + 45*A*a*b^2)*x^2 + 3*(7*B*a^3 + 15*A*a^2*b)*x)*e^4 + 2*
(88*B*b^3*d*x^3 - 142*B*a*b^2*d*x^2 - 23*B*a^2*b*d*x - 3*B*a^3*d)*e^3 + 14*(29*B*b^3*d^2*x^2 - 17*B*a*b^2*d^2*
x - B*a^2*b*d^2)*e^2 + 70*(5*B*b^3*d^3*x - B*a*b^2*d^3)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(b*d^5*e^4 - a*x^4*e^9
+ (b*d*x^4 - 4*a*d*x^3)*e^8 + 2*(2*b*d^2*x^3 - 3*a*d^2*x^2)*e^7 + 2*(3*b*d^3*x^2 - 2*a*d^3*x)*e^6 + (4*b*d^4*
x - a*d^4)*e^5)]

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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((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (134) = 268\).
time = 1.89, size = 635, normalized size = 3.80 \begin {gather*} -2 \, B b^{\frac {3}{2}} {\left | b \right |} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right ) - \frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left (\frac {{\left (176 \, B b^{10} d^{3} {\left | b \right |} e^{6} - 513 \, B a b^{9} d^{2} {\left | b \right |} e^{7} - 15 \, A b^{10} d^{2} {\left | b \right |} e^{7} + 498 \, B a^{2} b^{8} d {\left | b \right |} e^{8} + 30 \, A a b^{9} d {\left | b \right |} e^{8} - 161 \, B a^{3} b^{7} {\left | b \right |} e^{9} - 15 \, A a^{2} b^{8} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}} + \frac {406 \, {\left (B b^{11} d^{4} {\left | b \right |} e^{5} - 4 \, B a b^{10} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{2} b^{9} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{3} b^{8} d {\left | b \right |} e^{8} + B a^{4} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} + \frac {350 \, {\left (B b^{12} d^{5} {\left | b \right |} e^{4} - 5 \, B a b^{11} d^{4} {\left | b \right |} e^{5} + 10 \, B a^{2} b^{10} d^{3} {\left | b \right |} e^{6} - 10 \, B a^{3} b^{9} d^{2} {\left | b \right |} e^{7} + 5 \, B a^{4} b^{8} d {\left | b \right |} e^{8} - B a^{5} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} {\left (b x + a\right )} + \frac {105 \, {\left (B b^{13} d^{6} {\left | b \right |} e^{3} - 6 \, B a b^{12} d^{5} {\left | b \right |} e^{4} + 15 \, B a^{2} b^{11} d^{4} {\left | b \right |} e^{5} - 20 \, B a^{3} b^{10} d^{3} {\left | b \right |} e^{6} + 15 \, B a^{4} b^{9} d^{2} {\left | b \right |} e^{7} - 6 \, B a^{5} b^{8} d {\left | b \right |} e^{8} + B a^{6} b^{7} {\left | b \right |} e^{9}\right )}}{b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-2*B*b^(3/2)*abs(b)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e))) -
2/105*(((b*x + a)*((176*B*b^10*d^3*abs(b)*e^6 - 513*B*a*b^9*d^2*abs(b)*e^7 - 15*A*b^10*d^2*abs(b)*e^7 + 498*B*
a^2*b^8*d*abs(b)*e^8 + 30*A*a*b^9*d*abs(b)*e^8 - 161*B*a^3*b^7*abs(b)*e^9 - 15*A*a^2*b^8*abs(b)*e^9)*(b*x + a)
/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10) + 406*(B*b^11*d^4*abs(b)*e^5 - 4*B*a*b^10*d^
3*abs(b)*e^6 + 6*B*a^2*b^9*d^2*abs(b)*e^7 - 4*B*a^3*b^8*d*abs(b)*e^8 + B*a^4*b^7*abs(b)*e^9)/(b^5*d^3*e^7 - 3*
a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)) + 350*(B*b^12*d^5*abs(b)*e^4 - 5*B*a*b^11*d^4*abs(b)*e^5 + 10
*B*a^2*b^10*d^3*abs(b)*e^6 - 10*B*a^3*b^9*d^2*abs(b)*e^7 + 5*B*a^4*b^8*d*abs(b)*e^8 - B*a^5*b^7*abs(b)*e^9)/(b
^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10))*(b*x + a) + 105*(B*b^13*d^6*abs(b)*e^3 - 6*B*a
*b^12*d^5*abs(b)*e^4 + 15*B*a^2*b^11*d^4*abs(b)*e^5 - 20*B*a^3*b^10*d^3*abs(b)*e^6 + 15*B*a^4*b^9*d^2*abs(b)*e
^7 - 6*B*a^5*b^8*d*abs(b)*e^8 + B*a^6*b^7*abs(b)*e^9)/(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b
^2*e^10))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2),x)

[Out]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(9/2), x)

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