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

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

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Rubi [A]  time = 0.21, 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 = {78, 50, 63, 217, 206} \begin {gather*} \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 {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 {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 (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 {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 50

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 63

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 78

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] || LtQ
[p, n]))))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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 ) \operatorname {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 ) \operatorname {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 = 1.59, size = 309, normalized size = 1.22 \begin {gather*} \frac {\frac {\sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}} (a B e+6 A b e-7 b B d) \left (8 b^3 e^3 (a+b x)^3 \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 b^3 \sqrt {e} \sqrt {a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )}{3 b^4}+16 e^4 (a+b x)^4 (B d-A e)}{8 e^5 \sqrt {a+b x} \sqrt {d+e x} (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*e^4*(B*d - A*e)*(a + b*x)^4 + (Sqrt[b*d - a*e]*(-7*b*B*d + 6*A*b*e + a*B*e)*Sqrt[(b*(d + e*x))/(b*d - a*e)
]*(15*b^3*e*(b*d - a*e)^(5/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*b^3*e^2*(b*d - a*e)^(3/2)*(a + b*
x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^3*e^3*Sqrt[b*d - a*e]*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d - a*e)] -
15*b^3*Sqrt[e]*(b*d - a*e)^3*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/(3*b^4))/(8*e^5
*(-(b*d) + a*e)*Sqrt[a + b*x]*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.78, size = 315, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {a+b x} (a e-b d)^2 \left (\frac {240 A b^2 e^2 (a+b x)}{d+e x}+\frac {48 A e^4 (a+b x)^3}{(d+e x)^3}-\frac {198 A b e^3 (a+b x)^2}{(d+e x)^2}-\frac {280 b^2 B d e (a+b x)}{d+e x}-15 a b^2 B e-\frac {33 a B e^3 (a+b x)^2}{(d+e x)^2}-\frac {48 B d e^3 (a+b x)^3}{(d+e x)^3}+\frac {40 a b B e^2 (a+b x)}{d+e x}+\frac {231 b B d e^2 (a+b x)^2}{(d+e x)^2}-90 A b^3 e+105 b^3 B d\right )}{24 e^4 \sqrt {d+e x} \left (\frac {e (a+b x)}{d+e x}-b\right )^3}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*((-(b*d) + a*e)^2*Sqrt[a + b*x]*(105*b^3*B*d - 90*A*b^3*e - 15*a*b^2*B*e - (48*B*d*e^3*(a + b*x)^3)/(d +
e*x)^3 + (48*A*e^4*(a + b*x)^3)/(d + e*x)^3 + (231*b*B*d*e^2*(a + b*x)^2)/(d + e*x)^2 - (198*A*b*e^3*(a + b*x
)^2)/(d + e*x)^2 - (33*a*B*e^3*(a + b*x)^2)/(d + e*x)^2 - (280*b^2*B*d*e*(a + b*x))/(d + e*x) + (240*A*b^2*e^2
*(a + b*x))/(d + e*x) + (40*a*b*B*e^2*(a + b*x))/(d + e*x)))/(e^4*Sqrt[d + e*x]*(-b + (e*(a + b*x))/(d + e*x))
^3) - (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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fricas [A]  time = 4.56, size = 858, normalized size = 3.39 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{3} d^{4} - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3} e + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 6 \, A a^{2} b\right )} d e^{3} + {\left (7 \, B b^{3} d^{3} e - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 6 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (8 \, B b^{3} e^{4} x^{3} + 105 \, B b^{3} d^{3} e - 48 \, A a^{2} b e^{4} - 10 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d e^{3} - 2 \, {\left (7 \, B b^{3} d e^{3} - {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} e^{4}\right )} x^{2} + {\left (35 \, B b^{3} d^{2} e^{2} - 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d e^{3} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, {\left (b e^{6} x + b d e^{5}\right )}}, \frac {15 \, {\left (7 \, B b^{3} d^{4} - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{3} e + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 6 \, A a^{2} b\right )} d e^{3} + {\left (7 \, B b^{3} d^{3} e - 3 \, {\left (5 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 6 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, B b^{3} e^{4} x^{3} + 105 \, B b^{3} d^{3} e - 48 \, A a^{2} b e^{4} - 10 \, {\left (19 \, B a b^{2} + 9 \, A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (27 \, B a^{2} b + 50 \, A a b^{2}\right )} d e^{3} - 2 \, {\left (7 \, B b^{3} d e^{3} - {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} e^{4}\right )} x^{2} + {\left (35 \, B b^{3} d^{2} e^{2} - 2 \, {\left (34 \, B a b^{2} + 15 \, A b^{3}\right )} d e^{3} + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (b e^{6} x + 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 - 3*(5*B*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2
*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*
a^2*b)*e^4)*x)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)
*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(8*B*b^3*e^4*x^3 + 105*B*b^3*d^3*e - 48*A*a^2*b*e^
4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2*(7*B*b^3*d*e^3 - (13*B*a*b^2 + 6
*A*b^3)*e^4)*x^2 + (35*B*b^3*d^2*e^2 - 2*(34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a^2*b + 18*A*a*b^2)*e^4)*x)*s
qrt(b*x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5), 1/48*(15*(7*B*b^3*d^4 - 3*(5*B*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*
B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 +
3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*a^2*b)*e^4)*x)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt
(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(8*B*b^3*e^4*x^3 + 105
*B*b^3*d^3*e - 48*A*a^2*b*e^4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2*(7*B
*b^3*d*e^3 - (13*B*a*b^2 + 6*A*b^3)*e^4)*x^2 + (35*B*b^3*d^2*e^2 - 2*(34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a
^2*b + 18*A*a*b^2)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5)]

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giac [A]  time = 2.10, 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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maple [B]  time = 0.03, size = 1184, normalized size = 4.68

result too large to display

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)

[Out]

1/48*(b*x+a)^(1/2)*(-96*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a^2*e^3-180*A*b^2*d^2*e*((b*x+a)*(e*x+d))^(1/2)*
(b*e)^(1/2)+90*A*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x*a^2*b*e^4+90*A*
ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x*b^3*d^2*e^2+16*B*x^3*b^2*e^3*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*A*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/
2))*a^2*b*d*e^3-180*A*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a*b^2*d^2*e^
2-135*B*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^2*b*d^2*e^2-105*B*b^3*d^
4*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+225*B*a*b^2*d^2*e^2*x*ln(1/2*(2*
b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+52*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b
*e^3*x^2-28*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d*e^2*x^2+108*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*a*b*
e^3*x+70*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d^2*e*x-380*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b*d^2*e
-136*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a*b*d*e^2*x+210*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*b^2*d^3+15*B*
ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*x*a^3*e^4+90*A*ln(1/2*(2*b*e*x+a*e
+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*b^3*d^3*e+15*B*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))*a^3*d*e^3-60*A*x*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+300*A*a*b
*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*A*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
)/(b*e)^(1/2))*x*a*b^2*d*e^3-135*B*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))
*x*a^2*b*d*e^3+24*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*A*b^2*e^3*x^2-105*B*b^3*d^3*e*x*ln(1/2*(2*b*e*x+a*e+b*d+
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+225*B*a*b^2*d^3*e*ln(1/2*(2*b*e*x+a*e+b*d+2*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+66*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*B*a^2*e^3*x+162*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)*B*a^2*d*e^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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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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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sympy [F(-1)]  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)**(3/2),x)

[Out]

Timed out

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