3.16.99 \(\int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=171 \[ -\frac {2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac {6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac {6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[d + e*x])/e^5 - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(3/2)
)/(3*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*
a*B*e)*(d + e*x)^(7/2))/(7*e^5) + (2*b^3*B*(d + e*x)^(9/2))/(9*e^5)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 \sqrt {d+e x}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) \sqrt {d+e x}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{5/2}}{e^4}+\frac {b^3 B (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) \sqrt {d+e x}}{e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2}}{3 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^3 B (d+e x)^{9/2}}{9 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 145, normalized size = 0.85 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-45 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+189 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-105 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+315 (b d-a e)^3 (B d-A e)+35 b^3 B (d+e x)^4\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*(b*d - a*e)^3*(B*d - A*e) - 105*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x) + 18
9*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 45*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3 + 35*b^
3*B*(d + e*x)^4))/(315*e^5)

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IntegrateAlgebraic [B]  time = 0.14, size = 346, normalized size = 2.02 \begin {gather*} \frac {2 \sqrt {d+e x} \left (315 a^3 A e^4+105 a^3 B e^3 (d+e x)-315 a^3 B d e^3+315 a^2 A b e^3 (d+e x)-945 a^2 A b d e^3+945 a^2 b B d^2 e^2-630 a^2 b B d e^2 (d+e x)+189 a^2 b B e^2 (d+e x)^2+945 a A b^2 d^2 e^2-630 a A b^2 d e^2 (d+e x)+189 a A b^2 e^2 (d+e x)^2-945 a b^2 B d^3 e+945 a b^2 B d^2 e (d+e x)-567 a b^2 B d e (d+e x)^2+135 a b^2 B e (d+e x)^3-315 A b^3 d^3 e+315 A b^3 d^2 e (d+e x)-189 A b^3 d e (d+e x)^2+45 A b^3 e (d+e x)^3+315 b^3 B d^4-420 b^3 B d^3 (d+e x)+378 b^3 B d^2 (d+e x)^2-180 b^3 B d (d+e x)^3+35 b^3 B (d+e x)^4\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*b^3*B*d^4 - 315*A*b^3*d^3*e - 945*a*b^2*B*d^3*e + 945*a*A*b^2*d^2*e^2 + 945*a^2*b*B*d^2*
e^2 - 945*a^2*A*b*d*e^3 - 315*a^3*B*d*e^3 + 315*a^3*A*e^4 - 420*b^3*B*d^3*(d + e*x) + 315*A*b^3*d^2*e*(d + e*x
) + 945*a*b^2*B*d^2*e*(d + e*x) - 630*a*A*b^2*d*e^2*(d + e*x) - 630*a^2*b*B*d*e^2*(d + e*x) + 315*a^2*A*b*e^3*
(d + e*x) + 105*a^3*B*e^3*(d + e*x) + 378*b^3*B*d^2*(d + e*x)^2 - 189*A*b^3*d*e*(d + e*x)^2 - 567*a*b^2*B*d*e*
(d + e*x)^2 + 189*a*A*b^2*e^2*(d + e*x)^2 + 189*a^2*b*B*e^2*(d + e*x)^2 - 180*b^3*B*d*(d + e*x)^3 + 45*A*b^3*e
*(d + e*x)^3 + 135*a*b^2*B*e*(d + e*x)^3 + 35*b^3*B*(d + e*x)^4))/(315*e^5)

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fricas [A]  time = 1.09, size = 263, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 9 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{2} e^{2} - 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - {\left (64 \, B b^{3} d^{3} e - 72 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^3*e^4*x^4 + 128*B*b^3*d^4 + 315*A*a^3*e^4 - 144*(3*B*a*b^2 + A*b^3)*d^3*e + 504*(B*a^2*b + A*a*b
^2)*d^2*e^2 - 210*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d*e^3 - 9*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 3*(16*B*b^3*
d^2*e^2 - 18*(3*B*a*b^2 + A*b^3)*d*e^3 + 63*(B*a^2*b + A*a*b^2)*e^4)*x^2 - (64*B*b^3*d^3*e - 72*(3*B*a*b^2 + A
*b^3)*d^2*e^2 + 252*(B*a^2*b + A*a*b^2)*d*e^3 - 105*(B*a^3 + 3*A*a^2*b)*e^4)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 1.31, size = 345, normalized size = 2.02 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{3} e^{\left (-1\right )} + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} b e^{\left (-1\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} b e^{\left (-2\right )} + 63 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b^{2} e^{\left (-2\right )} + 27 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b^{2} e^{\left (-3\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{3} e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{3} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} A a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1) + 315*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^
2*b*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*b*e^(-2) + 63*(3*(x*e
+ d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*e^(-2) + 27*(5*(x*e + d)^(7/2) - 21*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a*b^2*e^(-3) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b^3*e^(-3) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^
(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b^3*e^(-4) + 315*sqrt(x
*e + d)*A*a^3)*e^(-1)

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maple [A]  time = 0.01, size = 301, normalized size = 1.76 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (35 B \,b^{3} x^{4} e^{4}+45 A \,b^{3} e^{4} x^{3}+135 B a \,b^{2} e^{4} x^{3}-40 B \,b^{3} d \,e^{3} x^{3}+189 A a \,b^{2} e^{4} x^{2}-54 A \,b^{3} d \,e^{3} x^{2}+189 B \,a^{2} b \,e^{4} x^{2}-162 B a \,b^{2} d \,e^{3} x^{2}+48 B \,b^{3} d^{2} e^{2} x^{2}+315 A \,a^{2} b \,e^{4} x -252 A a \,b^{2} d \,e^{3} x +72 A \,b^{3} d^{2} e^{2} x +105 B \,a^{3} e^{4} x -252 B \,a^{2} b d \,e^{3} x +216 B a \,b^{2} d^{2} e^{2} x -64 B \,b^{3} d^{3} e x +315 a^{3} A \,e^{4}-630 A \,a^{2} b d \,e^{3}+504 A a \,b^{2} d^{2} e^{2}-144 A \,b^{3} d^{3} e -210 B \,a^{3} d \,e^{3}+504 B \,a^{2} b \,d^{2} e^{2}-432 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right )}{315 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x)

[Out]

2/315*(e*x+d)^(1/2)*(35*B*b^3*e^4*x^4+45*A*b^3*e^4*x^3+135*B*a*b^2*e^4*x^3-40*B*b^3*d*e^3*x^3+189*A*a*b^2*e^4*
x^2-54*A*b^3*d*e^3*x^2+189*B*a^2*b*e^4*x^2-162*B*a*b^2*d*e^3*x^2+48*B*b^3*d^2*e^2*x^2+315*A*a^2*b*e^4*x-252*A*
a*b^2*d*e^3*x+72*A*b^3*d^2*e^2*x+105*B*a^3*e^4*x-252*B*a^2*b*d*e^3*x+216*B*a*b^2*d^2*e^2*x-64*B*b^3*d^3*e*x+31
5*A*a^3*e^4-630*A*a^2*b*d*e^3+504*A*a*b^2*d^2*e^2-144*A*b^3*d^3*e-210*B*a^3*d*e^3+504*B*a^2*b*d^2*e^2-432*B*a*
b^2*d^3*e+128*B*b^3*d^4)/e^5

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maxima [A]  time = 0.48, size = 265, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{3} - 45 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \sqrt {e x + d}\right )}}{315 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*(e*x + d)^(9/2)*B*b^3 - 45*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(7/2) + 189*(2*B*b^3*d^2 -
(3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)*(e*x + d)^(5/2) - 105*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*
d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(3/2) + 315*(B*b^3*d^4 + A*a^3*e^4 -
(3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*sqrt(e*x + d))/e^5

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mupad [B]  time = 1.23, size = 154, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{7\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{3\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{5\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^(1/2),x)

[Out]

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

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sympy [A]  time = 88.34, size = 916, normalized size = 5.36 \begin {gather*} \begin {cases} \frac {- \frac {2 A a^{3} d}{\sqrt {d + e x}} - 2 A a^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {6 A a^{2} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {6 A a^{2} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 A a b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {6 A a b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 A b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {2 A b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 B a^{3} d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a^{3} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 B a^{2} b d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {6 B a^{2} b \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {6 B a b^{2} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {6 B a b^{2} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 B b^{3} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 B b^{3} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + \frac {B b^{3} x^{5}}{5} + \frac {x^{4} \left (A b^{3} + 3 B a b^{2}\right )}{4} + \frac {x^{3} \left (3 A a b^{2} + 3 B a^{2} b\right )}{3} + \frac {x^{2} \left (3 A a^{2} b + B a^{3}\right )}{2}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*a**3*d/sqrt(d + e*x) - 2*A*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*A*a**2*b*d*(-d/sqrt(d
+ e*x) - sqrt(d + e*x))/e - 6*A*a**2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*A*a
*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*A*a*b**2*(-d**3/sqrt(d + e*x) -
3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 2*A*b**3*d*(-d**3/sqrt(d + e*x) - 3*d*
*2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 2*A*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(
d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*B*a**3*d*(-d/sqrt(d
+ e*x) - sqrt(d + e*x))/e - 2*B*a**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*B*a*
*2*b*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*B*a**2*b*(-d**3/sqrt(d + e*x) -
3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 6*B*a*b**2*d*(-d**3/sqrt(d + e*x) - 3*d
**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 6*B*a*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sq
rt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*B*b**3*d*(d**4/s
qrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e
**4 - 2*B*b**3*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5
/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((A*a**3*x + B*b**3*x**5/5 + x**4*(A*b*
*3 + 3*B*a*b**2)/4 + x**3*(3*A*a*b**2 + 3*B*a**2*b)/3 + x**2*(3*A*a**2*b + B*a**3)/2)/sqrt(d), True))

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