3.23.27 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx\) [2227]

Optimal. Leaf size=253 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {5 (b d-a e)^2 (7 b B d-6 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}} \]

[Out]

-5/8*(-a*e+b*d)^2*(-6*A*b*e-B*a*e+7*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/e^(9/2)/b^(1/2
)-2*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(1/2)-5/12*(-6*A*b*e-B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1
/2)/e^3+1/3*(-6*A*b*e-B*a*e+7*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/e^2/(-a*e+b*d)+5/8*(-a*e+b*d)*(-6*A*b*e-B*a*e
+7*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^4

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Rubi [A]
time = 0.13, antiderivative size = 253, 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, 52, 65, 223, 212} \begin {gather*} -\frac {5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}}+\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + (5*(b*d - a*e)*(7*b*B*d - 6*A*b*e - a*B*e)*Sq
rt[a + b*x]*Sqrt[d + e*x])/(8*e^4) - (5*(7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(12*e^3) +
((7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e^2*(b*d - a*e)) - (5*(b*d - a*e)^2*(7*b*B*d -
6*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*Sqrt[b]*e^(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 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)^{3/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(7 b B d-6 A b e-a B e) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {(5 (7 b B d-6 A b e-a B e)) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{6 e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}+\frac {(5 (b d-a e) (7 b B d-6 A b e-a B e)) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{8 e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\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 )}{8 b e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\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 )}{8 b e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 e^4}-\frac {5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 e^3}+\frac {(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt {d+e x}}{3 e^2 (b d-a e)}-\frac {5 (b d-a e)^2 (7 b B d-6 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 217, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x} \left (3 a^2 e^2 (27 B d-16 A e+11 B e x)+2 a b e \left (3 A e (25 d+9 e x)+B \left (-95 d^2-34 d e x+13 e^2 x^2\right )\right )+b^2 \left (6 A e \left (-15 d^2-5 d e x+2 e^2 x^2\right )+B \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )\right )}{24 e^4 \sqrt {d+e x}}+\frac {5 (b d-a e)^2 (-7 b B d+6 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 \sqrt {b} e^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(3*a^2*e^2*(27*B*d - 16*A*e + 11*B*e*x) + 2*a*b*e*(3*A*e*(25*d + 9*e*x) + B*(-95*d^2 - 34*d*e*x
+ 13*e^2*x^2)) + b^2*(6*A*e*(-15*d^2 - 5*d*e*x + 2*e^2*x^2) + B*(105*d^3 + 35*d^2*e*x - 14*d*e^2*x^2 + 8*e^3*
x^3))))/(24*e^4*Sqrt[d + e*x]) + (5*(b*d - a*e)^2*(-7*b*B*d + 6*A*b*e + a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])
/(Sqrt[b]*Sqrt[d + e*x])])/(8*Sqrt[b]*e^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1183\) vs. \(2(217)=434\).
time = 0.09, size = 1184, normalized size = 4.68

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (90 A \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^{2} b \,e^{4} x +90 A \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^{3} d^{2} e^{2} x +90 A \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^{2} b d \,e^{3}-180 A \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^{2} d^{2} e^{2}-135 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^{2} b \,d^{2} e^{2}+16 B \,b^{2} e^{3} x^{3} \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-180 A \,b^{2} d^{2} e \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+24 A \,b^{2} e^{3} x^{2} \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 ) b^{3} d^{3} e x +225 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^{2} d^{3} e +66 B \,a^{2} e^{3} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+162 B \,a^{2} d \,e^{2} \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 ) b^{3} d^{4}-60 A \,b^{2} d \,e^{2} x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+300 A a b d \,e^{2} \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-180 A \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^{2} d \,e^{3} x -135 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^{2} b d \,e^{3} x -96 A \,a^{2} e^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+210 B \,b^{2} d^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-136 B a b d \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+15 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^{3} e^{4} x +90 A \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^{3} d^{3} e +15 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^{3} d \,e^{3}+225 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^{2} d^{2} e^{2} x +52 B a b \,e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-28 B \,b^{2} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+108 A a b \,e^{3} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+70 B \,b^{2} d^{2} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-380 B a b \,d^{2} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{48 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {e x +d}\, e^{4}}\)	\(1184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(b*x+a)^(1/2)*(90*A*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^2*b*e^4
*x+90*A*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^3*d^2*e^2*x+90*A*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^2*b*d*e^3-180*A*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^2*d^2*e^2-135*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^2*b*d^2*e^2+16*B*b^2*e^3*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-180*
A*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+24*A*b^2*e^3*x^2*((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))*b^3*d^3*e*x+225*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^2*d^3*e+66*B*a^2*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+162*B*a^2*d*e^2*((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))*b^3*d^4-60*A*b^2*d*e^2*x*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+300*A*a*b*d*e^2
*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-180*A*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^2*d*e^3*x-135*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^2*b
*d*e^3*x-96*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*B*b^2*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-13
6*B*a*b*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+15*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^3*e^4*x+90*A*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^3*d^3*e+15*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^3*d*e^3+225*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^2*d^2*e^2*x+52*B*a*b*e^3*x^2*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-28*B*b^2*d*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+108*A*a*b*e^3*x*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+70*B*b^2*d^2*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-380*B*a*b*d^2*e*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+d)^(1/2)/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)^(3/2),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 [A]
time = 1.65, size = 830, normalized size = 3.28 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{3} d^{4} - {\left (B a^{3} + 6 \, A a^{2} b\right )} x e^{4} + {\left (3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d x - {\left (B a^{3} + 6 \, A a^{2} b\right )} d\right )} e^{3} - 3 \, {\left ({\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} x - {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2}\right )} e^{2} + {\left (7 \, B b^{3} d^{3} x - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\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^{3} e + {\left (8 \, B b^{3} x^{3} - 48 \, A a^{2} b + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{2} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x\right )} e^{4} - {\left (14 \, B b^{3} d x^{2} + 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d x - 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d\right )} e^{3} + 5 \, {\left (7 \, B b^{3} d^{2} x - 2 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{96 \, {\left (b x e^{6} + b d e^{5}\right )}}, \frac {15 \, {\left (7 \, B b^{3} d^{4} - {\left (B a^{3} + 6 \, A a^{2} b\right )} x e^{4} + {\left (3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d x - {\left (B a^{3} + 6 \, A a^{2} b\right )} d\right )} e^{3} - 3 \, {\left ({\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} x - {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2}\right )} e^{2} + {\left (7 \, B b^{3} d^{3} x - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (105 \, B b^{3} d^{3} e + {\left (8 \, B b^{3} x^{3} - 48 \, A a^{2} b + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{2} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x\right )} e^{4} - {\left (14 \, B b^{3} d x^{2} + 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d x - 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d\right )} e^{3} + 5 \, {\left (7 \, B b^{3} d^{2} x - 2 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{48 \, {\left (b x e^{6} + b d 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)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*(7*B*b^3*d^4 - (B*a^3 + 6*A*a^2*b)*x*e^4 + (3*(3*B*a^2*b + 4*A*a*b^2)*d*x - (B*a^3 + 6*A*a^2*b)*d)*
e^3 - 3*((5*B*a*b^2 + 2*A*b^3)*d^2*x - (3*B*a^2*b + 4*A*a*b^2)*d^2)*e^2 + (7*B*b^3*d^3*x - 3*(5*B*a*b^2 + 2*A*
b^3)*d^3)*e)*sqrt(b)*e^(1/2)*log(b^2*d^2 + 4*(b*d + (2*b*x + a)*e)*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^3*e + (8*B*b^3*x^3 - 48*A*a^2
*b + 2*(13*B*a*b^2 + 6*A*b^3)*x^2 + 3*(11*B*a^2*b + 18*A*a*b^2)*x)*e^4 - (14*B*b^3*d*x^2 + 2*(34*B*a*b^2 + 15*
A*b^3)*d*x - 3*(27*B*a^2*b + 50*A*a*b^2)*d)*e^3 + 5*(7*B*b^3*d^2*x - 2*(19*B*a*b^2 + 9*A*b^3)*d^2)*e^2)*sqrt(b
*x + a)*sqrt(x*e + d))/(b*x*e^6 + b*d*e^5), 1/48*(15*(7*B*b^3*d^4 - (B*a^3 + 6*A*a^2*b)*x*e^4 + (3*(3*B*a^2*b
+ 4*A*a*b^2)*d*x - (B*a^3 + 6*A*a^2*b)*d)*e^3 - 3*((5*B*a*b^2 + 2*A*b^3)*d^2*x - (3*B*a^2*b + 4*A*a*b^2)*d^2)*
e^2 + (7*B*b^3*d^3*x - 3*(5*B*a*b^2 + 2*A*b^3)*d^3)*e)*sqrt(-b*e)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x +
a)*sqrt(-b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*d*x + a*b*d)*e)) + 2*(105*B*b^3*d^3*e + (8*B*b^3*x^3
- 48*A*a^2*b + 2*(13*B*a*b^2 + 6*A*b^3)*x^2 + 3*(11*B*a^2*b + 18*A*a*b^2)*x)*e^4 - (14*B*b^3*d*x^2 + 2*(34*B*
a*b^2 + 15*A*b^3)*d*x - 3*(27*B*a^2*b + 50*A*a*b^2)*d)*e^3 + 5*(7*B*b^3*d^2*x - 2*(19*B*a*b^2 + 9*A*b^3)*d^2)*
e^2)*sqrt(b*x + a)*sqrt(x*e + d))/(b*x*e^6 + b*d*e^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(5/2)/(d + e*x)**(3/2), x)

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Giac [A]
time = 1.47, size = 396, normalized size = 1.57 \begin {gather*} \frac {5 \, {\left (7 \, B b^{3} d^{3} {\left | b \right |} - 15 \, B a b^{2} d^{2} {\left | b \right |} e - 6 \, A b^{3} d^{2} {\left | b \right |} e + 9 \, B a^{2} b d {\left | b \right |} e^{2} + 12 \, A a b^{2} d {\left | b \right |} e^{2} - B a^{3} {\left | b \right |} e^{3} - 6 \, A a^{2} b {\left | b \right |} e^{3}\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 )}{8 \, b^{\frac {3}{2}}} + \frac {{\left ({\left (2 \, {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |} e^{\left (-1\right )}}{b} - \frac {{\left (7 \, B b^{2} d {\left | b \right |} e^{5} - B a b {\left | b \right |} e^{6} - 6 \, A b^{2} {\left | b \right |} e^{6}\right )} e^{\left (-7\right )}}{b^{2}}\right )} {\left (b x + a\right )} + \frac {5 \, {\left (7 \, B b^{3} d^{2} {\left | b \right |} e^{4} - 8 \, B a b^{2} d {\left | b \right |} e^{5} - 6 \, A b^{3} d {\left | b \right |} e^{5} + B a^{2} b {\left | b \right |} e^{6} + 6 \, A a b^{2} {\left | b \right |} e^{6}\right )} e^{\left (-7\right )}}{b^{2}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (7 \, B b^{4} d^{3} {\left | b \right |} e^{3} - 15 \, B a b^{3} d^{2} {\left | b \right |} e^{4} - 6 \, A b^{4} d^{2} {\left | b \right |} e^{4} + 9 \, B a^{2} b^{2} d {\left | b \right |} e^{5} + 12 \, A a b^{3} d {\left | b \right |} e^{5} - B a^{3} b {\left | b \right |} e^{6} - 6 \, A a^{2} b^{2} {\left | b \right |} e^{6}\right )} e^{\left (-7\right )}}{b^{2}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} \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)^(3/2),x, algorithm="giac")

[Out]

5/8*(7*B*b^3*d^3*abs(b) - 15*B*a*b^2*d^2*abs(b)*e - 6*A*b^3*d^2*abs(b)*e + 9*B*a^2*b*d*abs(b)*e^2 + 12*A*a*b^2
*d*abs(b)*e^2 - B*a^3*abs(b)*e^3 - 6*A*a^2*b*abs(b)*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqr
t(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2) + 1/24*((2*(4*(b*x + a)*B*abs(b)*e^(-1)/b - (7*B*b^2*d*abs(b)*e^5 -
B*a*b*abs(b)*e^6 - 6*A*b^2*abs(b)*e^6)*e^(-7)/b^2)*(b*x + a) + 5*(7*B*b^3*d^2*abs(b)*e^4 - 8*B*a*b^2*d*abs(b)
*e^5 - 6*A*b^3*d*abs(b)*e^5 + B*a^2*b*abs(b)*e^6 + 6*A*a*b^2*abs(b)*e^6)*e^(-7)/b^2)*(b*x + a) + 15*(7*B*b^4*d
^3*abs(b)*e^3 - 15*B*a*b^3*d^2*abs(b)*e^4 - 6*A*b^4*d^2*abs(b)*e^4 + 9*B*a^2*b^2*d*abs(b)*e^5 + 12*A*a*b^3*d*a
bs(b)*e^5 - B*a^3*b*abs(b)*e^6 - 6*A*a^2*b^2*abs(b)*e^6)*e^(-7)/b^2)*sqrt(b*x + a)/sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)

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

[Out]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(3/2), x)

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