3.18.51 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^2} \, dx\) [1751]

Optimal. Leaf size=214 \[ \frac {(b d-a e) (2 b B d+5 A b e-7 a B e) \sqrt {d+e x}}{b^4}+\frac {(2 b B d+5 A b e-7 a B e) (d+e x)^{3/2}}{3 b^3}+\frac {(2 b B d+5 A b e-7 a B e) (d+e x)^{5/2}}{5 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{7/2}}{b (b d-a e) (a+b x)}-\frac {(b d-a e)^{3/2} (2 b B d+5 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 214} \begin {gather*} -\frac {(b d-a e)^{3/2} (-7 a B e+5 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}+\frac {\sqrt {d+e x} (b d-a e) (-7 a B e+5 A b e+2 b B d)}{b^4}+\frac {(d+e x)^{3/2} (-7 a B e+5 A b e+2 b B d)}{3 b^3}+\frac {(d+e x)^{5/2} (-7 a B e+5 A b e+2 b B d)}{5 b^2 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{b (a+b x) (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)*(2*b*B*d + 5*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/b^4 + ((2*b*B*d + 5*A*b*e - 7*a*B*e)*(d + e*x)^(3/2)
)/(3*b^3) + ((2*b*B*d + 5*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(5*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(7/2)
)/(b*(b*d - a*e)*(a + b*x)) - ((b*d - a*e)^(3/2)*(2*b*B*d + 5*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])
/Sqrt[b*d - a*e]])/b^(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 214

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

Rubi steps

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

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Mathematica [A]
time = 0.58, size = 210, normalized size = 0.98 \begin {gather*} \frac {\sqrt {d+e x} \left (-5 A b \left (15 a^2 e^2+10 a b e (-2 d+e x)+b^2 \left (3 d^2-14 d e x-2 e^2 x^2\right )\right )+B \left (105 a^3 e^2+10 a^2 b e (-17 d+7 e x)+a b^2 \left (61 d^2-118 d e x-14 e^2 x^2\right )+2 b^3 x \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )\right )}{15 b^4 (a+b x)}+\frac {(-b d+a e)^{3/2} (2 b B d+5 A b e-7 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-5*A*b*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(3*d^2 - 14*d*e*x - 2*e^2*x^2)) + B*(105*a^3*
e^2 + 10*a^2*b*e*(-17*d + 7*e*x) + a*b^2*(61*d^2 - 118*d*e*x - 14*e^2*x^2) + 2*b^3*x*(23*d^2 + 11*d*e*x + 3*e^
2*x^2))))/(15*b^4*(a + b*x)) + ((-(b*d) + a*e)^(3/2)*(2*b*B*d + 5*A*b*e - 7*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[-(b*d) + a*e]])/b^(9/2)

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Maple [A]
time = 0.12, size = 339, normalized size = 1.58

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B \,b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A a b \,e^{2} \sqrt {e x +d}-2 A \,b^{2} d e \sqrt {e x +d}-3 B \,a^{2} e^{2} \sqrt {e x +d}+4 B a b d e \sqrt {e x +d}-B \,b^{2} d^{2} \sqrt {e x +d}\right )}{b^{4}}+\frac {\frac {2 \left (-\frac {1}{2} A \,a^{2} b \,e^{3}+A a \,b^{2} d \,e^{2}-\frac {1}{2} A \,b^{3} d^{2} e +\frac {1}{2} B \,a^{3} e^{3}-B \,a^{2} b d \,e^{2}+\frac {1}{2} B a \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (5 A \,a^{2} b \,e^{3}-10 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e -7 B \,a^{3} e^{3}+16 B \,a^{2} b d \,e^{2}-11 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{b^{4}}\)	\(339\)
default	\(-\frac {2 \left (-\frac {b^{2} B \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B a b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B \,b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A a b \,e^{2} \sqrt {e x +d}-2 A \,b^{2} d e \sqrt {e x +d}-3 B \,a^{2} e^{2} \sqrt {e x +d}+4 B a b d e \sqrt {e x +d}-B \,b^{2} d^{2} \sqrt {e x +d}\right )}{b^{4}}+\frac {\frac {2 \left (-\frac {1}{2} A \,a^{2} b \,e^{3}+A a \,b^{2} d \,e^{2}-\frac {1}{2} A \,b^{3} d^{2} e +\frac {1}{2} B \,a^{3} e^{3}-B \,a^{2} b d \,e^{2}+\frac {1}{2} B a \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (5 A \,a^{2} b \,e^{3}-10 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e -7 B \,a^{3} e^{3}+16 B \,a^{2} b d \,e^{2}-11 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{b^{4}}\)	\(339\)
risch	\(-\frac {2 \left (-3 b^{2} B \,x^{2} e^{2}-5 A \,b^{2} e^{2} x +10 B a b \,e^{2} x -11 B \,b^{2} d e x +30 A a b \,e^{2}-35 A \,b^{2} d e -45 B \,a^{2} e^{2}+70 B a b d e -23 b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{15 b^{4}}-\frac {\sqrt {e x +d}\, A \,e^{3} a^{2}}{b^{3} \left (b e x +a e \right )}+\frac {2 \sqrt {e x +d}\, A \,e^{2} a d}{b^{2} \left (b e x +a e \right )}-\frac {\sqrt {e x +d}\, A e \,d^{2}}{b \left (b e x +a e \right )}+\frac {\sqrt {e x +d}\, B \,a^{3} e^{3}}{b^{4} \left (b e x +a e \right )}-\frac {2 \sqrt {e x +d}\, B \,a^{2} e^{2} d}{b^{3} \left (b e x +a e \right )}+\frac {\sqrt {e x +d}\, B a e \,d^{2}}{b^{2} \left (b e x +a e \right )}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,e^{3} a^{2}}{b^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {10 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,e^{2} a d}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A e \,d^{2}}{b \sqrt {\left (a e -b d \right ) b}}-\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{3} e^{3}}{b^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {16 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{2} e^{2} d}{b^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {11 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a e \,d^{2}}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,d^{3}}{b \sqrt {\left (a e -b d \right ) b}}\)	\(581\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b^4*(-1/5*b^2*B*(e*x+d)^(5/2)-1/3*A*b^2*e*(e*x+d)^(3/2)+2/3*B*a*b*e*(e*x+d)^(3/2)-1/3*B*b^2*d*(e*x+d)^(3/2)
+2*A*a*b*e^2*(e*x+d)^(1/2)-2*A*b^2*d*e*(e*x+d)^(1/2)-3*B*a^2*e^2*(e*x+d)^(1/2)+4*B*a*b*d*e*(e*x+d)^(1/2)-B*b^2
*d^2*(e*x+d)^(1/2))+2/b^4*((-1/2*A*a^2*b*e^3+A*a*b^2*d*e^2-1/2*A*b^3*d^2*e+1/2*B*a^3*e^3-B*a^2*b*d*e^2+1/2*B*a
*b^2*d^2*e)*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+1/2*(5*A*a^2*b*e^3-10*A*a*b^2*d*e^2+5*A*b^3*d^2*e-7*B*a^3*e^3+16
*B*a^2*b*d*e^2-11*B*a*b^2*d^2*e+2*B*b^3*d^3)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(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((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^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 = 1.08, size = 651, normalized size = 3.04 \begin {gather*} \left [-\frac {15 \, {\left (2 \, B b^{3} d^{2} x + 2 \, B a b^{2} d^{2} + {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} e^{2} - {\left ({\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d x + {\left (9 \, B a^{2} b - 5 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (46 \, B b^{3} d^{2} x + {\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{2} + {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} e^{2} + 2 \, {\left (11 \, B b^{3} d x^{2} - {\left (59 \, B a b^{2} - 35 \, A b^{3}\right )} d x - 5 \, {\left (17 \, B a^{2} b - 10 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {15 \, {\left (2 \, B b^{3} d^{2} x + 2 \, B a b^{2} d^{2} + {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} e^{2} - {\left ({\left (9 \, B a b^{2} - 5 \, A b^{3}\right )} d x + {\left (9 \, B a^{2} b - 5 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (46 \, B b^{3} d^{2} x + {\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{2} + {\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} e^{2} + 2 \, {\left (11 \, B b^{3} d x^{2} - {\left (59 \, B a b^{2} - 35 \, A b^{3}\right )} d x - 5 \, {\left (17 \, B a^{2} b - 10 \, A a b^{2}\right )} d\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30*(15*(2*B*b^3*d^2*x + 2*B*a*b^2*d^2 + (7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*e^2 - ((9*B*a*b^
2 - 5*A*b^3)*d*x + (9*B*a^2*b - 5*A*a*b^2)*d)*e)*sqrt((b*d - a*e)/b)*log((2*b*d + 2*sqrt(x*e + d)*b*sqrt((b*d
- a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(46*B*b^3*d^2*x + (61*B*a*b^2 - 15*A*b^3)*d^2 + (6*B*b^3*x^3 + 105*B*a
^3 - 75*A*a^2*b - 2*(7*B*a*b^2 - 5*A*b^3)*x^2 + 10*(7*B*a^2*b - 5*A*a*b^2)*x)*e^2 + 2*(11*B*b^3*d*x^2 - (59*B*
a*b^2 - 35*A*b^3)*d*x - 5*(17*B*a^2*b - 10*A*a*b^2)*d)*e)*sqrt(x*e + d))/(b^5*x + a*b^4), -1/15*(15*(2*B*b^3*d
^2*x + 2*B*a*b^2*d^2 + (7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*e^2 - ((9*B*a*b^2 - 5*A*b^3)*d*x + (9
*B*a^2*b - 5*A*a*b^2)*d)*e)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (
46*B*b^3*d^2*x + (61*B*a*b^2 - 15*A*b^3)*d^2 + (6*B*b^3*x^3 + 105*B*a^3 - 75*A*a^2*b - 2*(7*B*a*b^2 - 5*A*b^3)
*x^2 + 10*(7*B*a^2*b - 5*A*a*b^2)*x)*e^2 + 2*(11*B*b^3*d*x^2 - (59*B*a*b^2 - 35*A*b^3)*d*x - 5*(17*B*a^2*b - 1
0*A*a*b^2)*d)*e)*sqrt(x*e + d))/(b^5*x + a*b^4)]

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

[Out]

Timed out

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Giac [A]
time = 1.04, size = 400, normalized size = 1.87 \begin {gather*} \frac {{\left (2 \, B b^{3} d^{3} - 11 \, B a b^{2} d^{2} e + 5 \, A b^{3} d^{2} e + 16 \, B a^{2} b d e^{2} - 10 \, A a b^{2} d e^{2} - 7 \, B a^{3} e^{3} + 5 \, A a^{2} b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {\sqrt {x e + d} B a b^{2} d^{2} e - \sqrt {x e + d} A b^{3} d^{2} e - 2 \, \sqrt {x e + d} B a^{2} b d e^{2} + 2 \, \sqrt {x e + d} A a b^{2} d e^{2} + \sqrt {x e + d} B a^{3} e^{3} - \sqrt {x e + d} A a^{2} b e^{3}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{8} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{8} d + 15 \, \sqrt {x e + d} B b^{8} d^{2} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{7} e + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{8} e - 60 \, \sqrt {x e + d} B a b^{7} d e + 30 \, \sqrt {x e + d} A b^{8} d e + 45 \, \sqrt {x e + d} B a^{2} b^{6} e^{2} - 30 \, \sqrt {x e + d} A a b^{7} e^{2}\right )}}{15 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*b^3*d^3 - 11*B*a*b^2*d^2*e + 5*A*b^3*d^2*e + 16*B*a^2*b*d*e^2 - 10*A*a*b^2*d*e^2 - 7*B*a^3*e^3 + 5*A*a^2*
b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + (sqrt(x*e + d)*B*a*b^2*d^2*e
- sqrt(x*e + d)*A*b^3*d^2*e - 2*sqrt(x*e + d)*B*a^2*b*d*e^2 + 2*sqrt(x*e + d)*A*a*b^2*d*e^2 + sqrt(x*e + d)*B*
a^3*e^3 - sqrt(x*e + d)*A*a^2*b*e^3)/(((x*e + d)*b - b*d + a*e)*b^4) + 2/15*(3*(x*e + d)^(5/2)*B*b^8 + 5*(x*e
+ d)^(3/2)*B*b^8*d + 15*sqrt(x*e + d)*B*b^8*d^2 - 10*(x*e + d)^(3/2)*B*a*b^7*e + 5*(x*e + d)^(3/2)*A*b^8*e - 6
0*sqrt(x*e + d)*B*a*b^7*d*e + 30*sqrt(x*e + d)*A*b^8*d*e + 45*sqrt(x*e + d)*B*a^2*b^6*e^2 - 30*sqrt(x*e + d)*A
*a*b^7*e^2)/b^10

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Mupad [B]
time = 0.13, size = 363, normalized size = 1.70 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{3\,b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}+\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (B\,a^3\,e^3-2\,B\,a^2\,b\,d\,e^2-A\,a^2\,b\,e^3+B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2-A\,b^3\,d^2\,e\right )}{b^5\,\left (d+e\,x\right )-b^5\,d+a\,b^4\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{5/2}}{5\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e-7\,B\,a\,e+2\,B\,b\,d\right )}{-7\,B\,a^3\,e^3+16\,B\,a^2\,b\,d\,e^2+5\,A\,a^2\,b\,e^3-11\,B\,a\,b^2\,d^2\,e-10\,A\,a\,b^2\,d\,e^2+2\,B\,b^3\,d^3+5\,A\,b^3\,d^2\,e}\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\left (5\,A\,b\,e-7\,B\,a\,e+2\,B\,b\,d\right )}{b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*A*e - 2*B*d)/(3*b^2) + (2*B*(2*b^2*d - 2*a*b*e))/(3*b^4))*(d + e*x)^(3/2) + (((2*b^2*d - 2*a*b*e)*((2*A*e
- 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/b^2 - (2*B*(a*e - b*d)^2)/b^4)*(d + e*x)^(1/2) + ((d + e*x)^(1/
2)*(B*a^3*e^3 - A*a^2*b*e^3 - A*b^3*d^2*e + 2*A*a*b^2*d*e^2 + B*a*b^2*d^2*e - 2*B*a^2*b*d*e^2))/(b^5*(d + e*x)
- b^5*d + a*b^4*e) + (2*B*(d + e*x)^(5/2))/(5*b^2) + (atan((b^(1/2)*(a*e - b*d)^(3/2)*(d + e*x)^(1/2)*(5*A*b*
e - 7*B*a*e + 2*B*b*d))/(2*B*b^3*d^3 - 7*B*a^3*e^3 + 5*A*a^2*b*e^3 + 5*A*b^3*d^2*e - 10*A*a*b^2*d*e^2 - 11*B*a
*b^2*d^2*e + 16*B*a^2*b*d*e^2))*(a*e - b*d)^(3/2)*(5*A*b*e - 7*B*a*e + 2*B*b*d))/b^(9/2)

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