3.18.42 \(\int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx\) [1742]

Optimal. Leaf size=198 \[ \frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214} \begin {gather*} -\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac {2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac {2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (2*(A*b - a*B)*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*b^4) + (2*(
A*b - a*B)*(b*d - a*e)*(d + e*x)^(5/2))/(5*b^3) + (2*(A*b - a*B)*(d + e*x)^(7/2))/(7*b^2) + (2*B*(d + e*x)^(9/
2))/(9*b*e) - (2*(A*b - a*B)*(b*d - a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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 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)^{7/2}}{a+b x} \, dx &=\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 \left (\frac {9 A b e}{2}-\frac {9 a B e}{2}\right )\right ) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{9 b e}\\ &=\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{b^2}\\ &=\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{b^3}\\ &=\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^4}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^5}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 (A b-a B) (b d-a e)^4\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^5 e}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 263, normalized size = 1.33 \begin {gather*} \frac {2 \sqrt {d+e x} \left (315 a^4 B e^4-105 a^3 b e^3 (10 B d+3 A e+B e x)+21 a^2 b^2 e^2 \left (5 A e (10 d+e x)+B \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )-3 a b^3 e \left (7 A e \left (58 d^2+16 d e x+3 e^2 x^2\right )+B \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+b^4 \left (35 B (d+e x)^4+3 A e \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{315 b^5 e}+\frac {2 (A b-a B) (-b d+a e)^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x]

[Out]

(2*Sqrt[d + e*x]*(315*a^4*B*e^4 - 105*a^3*b*e^3*(10*B*d + 3*A*e + B*e*x) + 21*a^2*b^2*e^2*(5*A*e*(10*d + e*x)
+ B*(58*d^2 + 16*d*e*x + 3*e^2*x^2)) - 3*a*b^3*e*(7*A*e*(58*d^2 + 16*d*e*x + 3*e^2*x^2) + B*(176*d^3 + 122*d^2
*e*x + 66*d*e^2*x^2 + 15*e^3*x^3)) + b^4*(35*B*(d + e*x)^4 + 3*A*e*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*
e^3*x^3))))/(315*b^5*e) + (2*(A*b - a*B)*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e
]])/b^(11/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(530\) vs. \(2(170)=340\).
time = 0.10, size = 531, normalized size = 2.68 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2/e*(-1/b^5*(-1/9*B*(e*x+d)^(9/2)*b^4-1/7*A*b^4*e*(e*x+d)^(7/2)+1/7*B*a*b^3*e*(e*x+d)^(7/2)+1/5*A*a*b^3*e^2*(e
*x+d)^(5/2)-1/5*A*b^4*d*e*(e*x+d)^(5/2)-1/5*B*a^2*b^2*e^2*(e*x+d)^(5/2)+1/5*B*a*b^3*d*e*(e*x+d)^(5/2)-1/3*A*a^
2*b^2*e^3*(e*x+d)^(3/2)+2/3*A*a*b^3*d*e^2*(e*x+d)^(3/2)-1/3*A*b^4*d^2*e*(e*x+d)^(3/2)+1/3*B*a^3*b*e^3*(e*x+d)^
(3/2)-2/3*B*a^2*b^2*d*e^2*(e*x+d)^(3/2)+1/3*B*a*b^3*d^2*e*(e*x+d)^(3/2)+A*a^3*b*e^4*(e*x+d)^(1/2)-3*A*a^2*b^2*
d*e^3*(e*x+d)^(1/2)+3*A*a*b^3*d^2*e^2*(e*x+d)^(1/2)-A*b^4*d^3*e*(e*x+d)^(1/2)-B*a^4*e^4*(e*x+d)^(1/2)+3*B*a^3*
b*d*e^3*(e*x+d)^(1/2)-3*B*a^2*b^2*d^2*e^2*(e*x+d)^(1/2)+B*a*b^3*d^3*e*(e*x+d)^(1/2))+e*(A*a^4*b*e^4-4*A*a^3*b^
2*d*e^3+6*A*a^2*b^3*d^2*e^2-4*A*a*b^4*d^3*e+A*b^5*d^4-B*a^5*e^4+4*B*a^4*b*d*e^3-6*B*a^3*b^2*d^2*e^2+4*B*a^2*b^
3*d^3*e-B*a*b^4*d^4)/b^5/((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)^(7/2)/(b*x+a),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. 412 vs. \(2 (180) = 360\).
time = 1.18, size = 836, normalized size = 4.22 \begin {gather*} \left [-\frac {{\left (315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\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 (35 \, B b^{4} d^{4} + {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} e^{4} + 2 \, {\left (70 \, B b^{4} d x^{3} - 99 \, {\left (B a b^{3} - A b^{4}\right )} d x^{2} + 168 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d x - 525 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d\right )} e^{3} + 6 \, {\left (35 \, B b^{4} d^{2} x^{2} - 61 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} x + 203 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2}\right )} e^{2} + 4 \, {\left (35 \, B b^{4} d^{3} x - 132 \, {\left (B a b^{3} - A b^{4}\right )} d^{3}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{315 \, b^{5}}, \frac {2 \, {\left (315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\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 (35 \, B b^{4} d^{4} + {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} e^{4} + 2 \, {\left (70 \, B b^{4} d x^{3} - 99 \, {\left (B a b^{3} - A b^{4}\right )} d x^{2} + 168 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d x - 525 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d\right )} e^{3} + 6 \, {\left (35 \, B b^{4} d^{2} x^{2} - 61 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} x + 203 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2}\right )} e^{2} + 4 \, {\left (35 \, B b^{4} d^{3} x - 132 \, {\left (B a b^{3} - A b^{4}\right )} d^{3}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{315 \, b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x, algorithm="fricas")

[Out]

[-1/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2)*d*e^3 - (B*a
^4 - A*a^3*b)*e^4)*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*(35*B*b^4*d^4 + (35*B*b^4*x^4 + 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 -
A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*e^4 + 2*(70*B*b^4*d*x^3 - 99*(B*a*b^3 - A*b^4)*d*x^2 + 168*(B*a^2
*b^2 - A*a*b^3)*d*x - 525*(B*a^3*b - A*a^2*b^2)*d)*e^3 + 6*(35*B*b^4*d^2*x^2 - 61*(B*a*b^3 - A*b^4)*d^2*x + 20
3*(B*a^2*b^2 - A*a*b^3)*d^2)*e^2 + 4*(35*B*b^4*d^3*x - 132*(B*a*b^3 - A*b^4)*d^3)*e)*sqrt(x*e + d))*e^(-1)/b^5
, 2/315*(315*((B*a*b^3 - A*b^4)*d^3*e - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + 3*(B*a^3*b - A*a^2*b^2)*d*e^3 - (B*a
^4 - A*a^3*b)*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) + (35*B*b^4*
d^4 + (35*B*b^4*x^4 + 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*
(B*a^3*b - A*a^2*b^2)*x)*e^4 + 2*(70*B*b^4*d*x^3 - 99*(B*a*b^3 - A*b^4)*d*x^2 + 168*(B*a^2*b^2 - A*a*b^3)*d*x
- 525*(B*a^3*b - A*a^2*b^2)*d)*e^3 + 6*(35*B*b^4*d^2*x^2 - 61*(B*a*b^3 - A*b^4)*d^2*x + 203*(B*a^2*b^2 - A*a*b
^3)*d^2)*e^2 + 4*(35*B*b^4*d^3*x - 132*(B*a*b^3 - A*b^4)*d^3)*e)*sqrt(x*e + d))*e^(-1)/b^5]

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Sympy [A]
time = 56.05, size = 337, normalized size = 1.70 \begin {gather*} \frac {2 B \left (d + e x\right )^{\frac {9}{2}}}{9 b e} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A b - 2 B a\right )}{7 b^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{5 b^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 A a^{2} b e^{2} - 4 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} + 4 B a^{2} b d e - 2 B a b^{2} d^{2}\right )}{3 b^{4}} + \frac {\sqrt {d + e x} \left (- 2 A a^{3} b e^{3} + 6 A a^{2} b^{2} d e^{2} - 6 A a b^{3} d^{2} e + 2 A b^{4} d^{3} + 2 B a^{4} e^{3} - 6 B a^{3} b d e^{2} + 6 B a^{2} b^{2} d^{2} e - 2 B a b^{3} d^{3}\right )}{b^{5}} - \frac {2 \left (- A b + B a\right ) \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{6} \sqrt {\frac {a e - b d}{b}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a),x)

[Out]

2*B*(d + e*x)**(9/2)/(9*b*e) + (d + e*x)**(7/2)*(2*A*b - 2*B*a)/(7*b**2) + (d + e*x)**(5/2)*(-2*A*a*b*e + 2*A*
b**2*d + 2*B*a**2*e - 2*B*a*b*d)/(5*b**3) + (d + e*x)**(3/2)*(2*A*a**2*b*e**2 - 4*A*a*b**2*d*e + 2*A*b**3*d**2
- 2*B*a**3*e**2 + 4*B*a**2*b*d*e - 2*B*a*b**2*d**2)/(3*b**4) + sqrt(d + e*x)*(-2*A*a**3*b*e**3 + 6*A*a**2*b**
2*d*e**2 - 6*A*a*b**3*d**2*e + 2*A*b**4*d**3 + 2*B*a**4*e**3 - 6*B*a**3*b*d*e**2 + 6*B*a**2*b**2*d**2*e - 2*B*
a*b**3*d**3)/b**5 - 2*(-A*b + B*a)*(a*e - b*d)**4*atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b**6*sqrt((a*e - b*
d)/b))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (180) = 360\).
time = 0.91, size = 558, normalized size = 2.82 \begin {gather*} -\frac {2 \, {\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\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^{5}} + \frac {2 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{8} e^{8} - 45 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{7} e^{9} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{8} e^{9} - 63 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{7} d e^{9} + 63 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{8} d e^{9} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{7} d^{2} e^{9} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt {x e + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt {x e + d} A b^{8} d^{3} e^{9} + 63 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{6} e^{10} - 63 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{7} e^{10} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{6} d e^{10} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt {x e + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt {x e + d} A a b^{7} d^{2} e^{10} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{5} e^{11} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt {x e + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt {x e + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt {x e + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt {x e + d} A a^{3} b^{5} e^{12}\right )} e^{\left (-9\right )}}{315 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a),x, algorithm="giac")

[Out]

-2*(B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d^2*e^2
- 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A*a^4*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/
(sqrt(-b^2*d + a*b*e)*b^5) + 2/315*(35*(x*e + d)^(9/2)*B*b^8*e^8 - 45*(x*e + d)^(7/2)*B*a*b^7*e^9 + 45*(x*e +
d)^(7/2)*A*b^8*e^9 - 63*(x*e + d)^(5/2)*B*a*b^7*d*e^9 + 63*(x*e + d)^(5/2)*A*b^8*d*e^9 - 105*(x*e + d)^(3/2)*B
*a*b^7*d^2*e^9 + 105*(x*e + d)^(3/2)*A*b^8*d^2*e^9 - 315*sqrt(x*e + d)*B*a*b^7*d^3*e^9 + 315*sqrt(x*e + d)*A*b
^8*d^3*e^9 + 63*(x*e + d)^(5/2)*B*a^2*b^6*e^10 - 63*(x*e + d)^(5/2)*A*a*b^7*e^10 + 210*(x*e + d)^(3/2)*B*a^2*b
^6*d*e^10 - 210*(x*e + d)^(3/2)*A*a*b^7*d*e^10 + 945*sqrt(x*e + d)*B*a^2*b^6*d^2*e^10 - 945*sqrt(x*e + d)*A*a*
b^7*d^2*e^10 - 105*(x*e + d)^(3/2)*B*a^3*b^5*e^11 + 105*(x*e + d)^(3/2)*A*a^2*b^6*e^11 - 945*sqrt(x*e + d)*B*a
^3*b^5*d*e^11 + 945*sqrt(x*e + d)*A*a^2*b^6*d*e^11 + 315*sqrt(x*e + d)*B*a^4*b^4*e^12 - 315*sqrt(x*e + d)*A*a^
3*b^5*e^12)*e^(-9)/b^9

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Mupad [B]
time = 0.12, size = 426, normalized size = 2.15 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{7\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{7\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{7/2}+\frac {2\,B\,{\left (d+e\,x\right )}^{9/2}}{9\,b\,e}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}}+\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2\,e^2}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^3\,\sqrt {d+e\,x}}{b^3\,e^3}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a + b*x),x)

[Out]

((2*A*e - 2*B*d)/(7*b*e) - (2*B*(a*e^2 - b*d*e))/(7*b^2*e^2))*(d + e*x)^(7/2) + (2*B*(d + e*x)^(9/2))/(9*b*e)
+ (2*atan((b^(1/2)*(A*b - B*a)*(a*e - b*d)^(7/2)*(d + e*x)^(1/2))/(A*b^5*d^4 - B*a^5*e^4 + A*a^4*b*e^4 - B*a*b
^4*d^4 - 4*A*a^3*b^2*d*e^3 + 4*B*a^2*b^3*d^3*e + 6*A*a^2*b^3*d^2*e^2 - 6*B*a^3*b^2*d^2*e^2 - 4*A*a*b^4*d^3*e +
4*B*a^4*b*d*e^3))*(A*b - B*a)*(a*e - b*d)^(7/2))/b^(11/2) + (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(
b^2*e^2))*(a*e^2 - b*d*e)^2*(d + e*x)^(3/2))/(3*b^2*e^2) - (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(b^
2*e^2))*(a*e^2 - b*d*e)^3*(d + e*x)^(1/2))/(b^3*e^3) - (((2*A*e - 2*B*d)/(b*e) - (2*B*(a*e^2 - b*d*e))/(b^2*e^
2))*(a*e^2 - b*d*e)*(d + e*x)^(5/2))/(5*b*e)

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