3.17.23 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \]

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Rubi [A]  time = 0.19, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) - (2*(b*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (6*b*(b*d
- a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x)) - (2*b^2*(
4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^5*(a + b*x)) + (2*b^3*B*(d +
e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^5*(a + b*x))

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]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{7/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{7/2}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{9/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{13/2}}{e^4}+\frac {b^6 B (d+e x)^{15/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{9/2} \left (-7293 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+25245 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-9945 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+12155 (b d-a e)^3 (B d-A e)+6435 b^3 B (d+e x)^4\right )}{109395 e^5 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(9/2)*(12155*(b*d - a*e)^3*(B*d - A*e) - 9945*(b*d - a*e)^2*(4*b*B*d - 3*A*b*
e - a*B*e)*(d + e*x) + 25245*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 7293*b^2*(4*b*B*d - A*b*e -
3*a*B*e)*(d + e*x)^3 + 6435*b^3*B*(d + e*x)^4))/(109395*e^5*(a + b*x)^3)

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IntegrateAlgebraic [A]  time = 53.77, size = 374, normalized size = 1.21 \begin {gather*} \frac {2 (d+e x)^{9/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (12155 a^3 A e^4+9945 a^3 B e^3 (d+e x)-12155 a^3 B d e^3+29835 a^2 A b e^3 (d+e x)-36465 a^2 A b d e^3+36465 a^2 b B d^2 e^2-59670 a^2 b B d e^2 (d+e x)+25245 a^2 b B e^2 (d+e x)^2+36465 a A b^2 d^2 e^2-59670 a A b^2 d e^2 (d+e x)+25245 a A b^2 e^2 (d+e x)^2-36465 a b^2 B d^3 e+89505 a b^2 B d^2 e (d+e x)-75735 a b^2 B d e (d+e x)^2+21879 a b^2 B e (d+e x)^3-12155 A b^3 d^3 e+29835 A b^3 d^2 e (d+e x)-25245 A b^3 d e (d+e x)^2+7293 A b^3 e (d+e x)^3+12155 b^3 B d^4-39780 b^3 B d^3 (d+e x)+50490 b^3 B d^2 (d+e x)^2-29172 b^3 B d (d+e x)^3+6435 b^3 B (d+e x)^4\right )}{109395 e^4 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(d + e*x)^(9/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(12155*b^3*B*d^4 - 12155*A*b^3*d^3*e - 36465*a*b^2*B*d^3*e + 3646
5*a*A*b^2*d^2*e^2 + 36465*a^2*b*B*d^2*e^2 - 36465*a^2*A*b*d*e^3 - 12155*a^3*B*d*e^3 + 12155*a^3*A*e^4 - 39780*
b^3*B*d^3*(d + e*x) + 29835*A*b^3*d^2*e*(d + e*x) + 89505*a*b^2*B*d^2*e*(d + e*x) - 59670*a*A*b^2*d*e^2*(d + e
*x) - 59670*a^2*b*B*d*e^2*(d + e*x) + 29835*a^2*A*b*e^3*(d + e*x) + 9945*a^3*B*e^3*(d + e*x) + 50490*b^3*B*d^2
*(d + e*x)^2 - 25245*A*b^3*d*e*(d + e*x)^2 - 75735*a*b^2*B*d*e*(d + e*x)^2 + 25245*a*A*b^2*e^2*(d + e*x)^2 + 2
5245*a^2*b*B*e^2*(d + e*x)^2 - 29172*b^3*B*d*(d + e*x)^3 + 7293*A*b^3*e*(d + e*x)^3 + 21879*a*b^2*B*e*(d + e*x
)^3 + 6435*b^3*B*(d + e*x)^4))/(109395*e^4*(a*e + b*e*x))

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fricas [B]  time = 0.44, size = 633, normalized size = 2.06 \begin {gather*} \frac {2 \, {\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \, {\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \, {\left (52 \, B b^{3} d e^{7} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \, {\left (802 \, B b^{3} d^{2} e^{6} + 782 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \, {\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \, {\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \, {\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*(3*B*a*b^2 + A*b^3)*d^7*e + 2040*(B*a
^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)
*x^7 + 33*(802*B*b^3*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*(1212*B*b^
3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2*e^6 + 10200*(B*a^2*b + A*a*b^2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)
*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*
e^6 + 7514*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*B*a*b^2 + A*b^3)*d^4
*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*A*a^2*b)*d^2*e^6)*x^3 + 3*(16*B*b^3*d^6*e^2 + 2431
0*A*a^3*d^2*e^6 - 34*(3*B*a*b^2 + A*b^3)*d^5*e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*
d^3*e^5)*x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d^6*e^2 + 1020*(B*a^2*b + A*a*b
^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.54, size = 3088, normalized size = 10.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

2/765765*(255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d^4*e^(-1)*sgn(b*x + a) + 765765*((x*e + d)^(3/2)
- 3*sqrt(x*e + d)*d)*A*a^2*b*d^4*e^(-1)*sgn(b*x + a) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*B*a^2*b*d^4*e^(-2)*sgn(b*x + a) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqr
t(x*e + d)*d^2)*A*a*b^2*d^4*e^(-2)*sgn(b*x + a) + 65637*(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*d^4*e^(-3)*sgn(b*x + a) + 21879*(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*d^4*e^(-3)*sgn(b*x + a) + 2431*(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*d^4*e^(-4)*sgn(b*x + a) + 204204*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*d^3
*e^(-1)*sgn(b*x + a) + 612612*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*d^3*e^
(-1)*sgn(b*x + a) + 262548*(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^2*b*d^3*e^(-2)*sgn(b*x + a) + 262548*(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*a*b^2*d^3*e^(-2)*sgn(b*x + a) + 29172*(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*a*b^2*d^3*e^(-3)*sgn(b*x +
a) + 9724*(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 + 31
5*sqrt(x*e + d)*d^4)*A*b^3*d^3*e^(-3)*sgn(b*x + a) + 4420*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(
x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*b^3*d^3*e^
(-4)*sgn(b*x + a) + 765765*sqrt(x*e + d)*A*a^3*d^4*sgn(b*x + a) + 1021020*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d
)*A*a^3*d^3*sgn(b*x + a) + 131274*(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^3*d^2*e^(-1)*sgn(b*x + a) + 393822*(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*a^2*b*d^2*e^(-1)*sgn(b*x + a) + 43758*(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*a^2*b*d^2*e^(-2)*sgn(b
*x + a) + 43758*(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)*A*a*b^2*d^2*e^(-2)*sgn(b*x + a) + 19890*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*
d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a
*b^2*d^2*e^(-3)*sgn(b*x + a) + 6630*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1
386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^3*d^2*e^(-3)*sgn(b*x + a) + 15
30*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 900
9*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^3*d^2*e^(-4)*sgn(b*x + a) + 306
306*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*d^2*sgn(b*x + a) + 9724*(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*a^3*d*e^(-1)*sgn(b*x + a) + 29172*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 4
20*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b*d*e^(-1)*sgn(b*x + a) + 13260*(63*(x*e + d)^(11/2) - 3
85*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqr
t(x*e + d)*d^5)*B*a^2*b*d*e^(-2)*sgn(b*x + a) + 13260*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e
+ d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a*b^2*d*e^(-2)
*sgn(b*x + a) + 3060*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e +
d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a*b^2*d*e^(-3)*
sgn(b*x + a) + 1020*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d
)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^3*d*e^(-3)*sgn
(b*x + a) + 476*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)
^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e
+ d)*d^7)*B*b^3*d*e^(-4)*sgn(b*x + a) + 87516*(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*a^3*d*sgn(b*x + a) + 1105*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*
e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^3*e^(-1)*s
gn(b*x + a) + 3315*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/
2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a^2*b*e^(-1)*sgn(b*x + a) + 765*(231*(x*e + d)^(1
3/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^
4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a^2*b*e^(-2)*sgn(b*x + a) + 765*(231*(x*e + d)^(13/2)
- 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 -
6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*a*b^2*e^(-2)*sgn(b*x + a) + 357*(429*(x*e + d)^(15/2) - 3
465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 -
27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*a*b^2*e^(-3)*sgn(b*x + a) +
119*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 +
32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*A
*b^3*e^(-3)*sgn(b*x + a) + 7*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 -
556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2
)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*b^3*e^(-4)*sgn(b*x + a) + 2431*(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)*A*a
^3*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.05, size = 317, normalized size = 1.03 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 b^{3} B \,x^{4} e^{4}+7293 A \,b^{3} e^{4} x^{3}+21879 B a \,b^{2} e^{4} x^{3}-3432 B \,b^{3} d \,e^{3} x^{3}+25245 A a \,b^{2} e^{4} x^{2}-3366 A \,b^{3} d \,e^{3} x^{2}+25245 B \,a^{2} b \,e^{4} x^{2}-10098 B a \,b^{2} d \,e^{3} x^{2}+1584 B \,b^{3} d^{2} e^{2} x^{2}+29835 A \,a^{2} b \,e^{4} x -9180 A a \,b^{2} d \,e^{3} x +1224 A \,b^{3} d^{2} e^{2} x +9945 B \,a^{3} e^{4} x -9180 B \,a^{2} b d \,e^{3} x +3672 B a \,b^{2} d^{2} e^{2} x -576 B \,b^{3} d^{3} e x +12155 A \,a^{3} e^{4}-6630 A \,a^{2} b d \,e^{3}+2040 A a \,b^{2} d^{2} e^{2}-272 A \,b^{3} d^{3} e -2210 B \,a^{3} d \,e^{3}+2040 B \,a^{2} b \,d^{2} e^{2}-816 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{109395 \left (b x +a \right )^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^2*e^4*x^3-3432*B*b^3*d*e^3*x^3+25245
*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+25245*B*a^2*b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+298
35*A*a^2*b*e^4*x-9180*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x-9180*B*a^2*b*d*e^3*x+3672*B*a*b^2*
d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3*e^4-6630*A*a^2*b*d*e^3+2040*A*a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3
*d*e^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [B]  time = 0.75, size = 697, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \, {\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \, {\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \, {\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \, {\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} A}{6435 \, e^{4}} + \frac {2 \, {\left (6435 \, b^{3} e^{8} x^{8} + 128 \, b^{3} d^{8} - 816 \, a b^{2} d^{7} e + 2040 \, a^{2} b d^{6} e^{2} - 2210 \, a^{3} d^{5} e^{3} + 429 \, {\left (52 \, b^{3} d e^{7} + 51 \, a b^{2} e^{8}\right )} x^{7} + 33 \, {\left (802 \, b^{3} d^{2} e^{6} + 2346 \, a b^{2} d e^{7} + 765 \, a^{2} b e^{8}\right )} x^{6} + 9 \, {\left (1212 \, b^{3} d^{3} e^{5} + 10506 \, a b^{2} d^{2} e^{6} + 10200 \, a^{2} b d e^{7} + 1105 \, a^{3} e^{8}\right )} x^{5} + 5 \, {\left (7 \, b^{3} d^{4} e^{4} + 8160 \, a b^{2} d^{3} e^{5} + 23358 \, a^{2} b d^{2} e^{6} + 7514 \, a^{3} d e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{5} e^{3} - 51 \, a b^{2} d^{4} e^{4} - 10812 \, a^{2} b d^{3} e^{5} - 10166 \, a^{3} d^{2} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{6} e^{2} - 102 \, a b^{2} d^{5} e^{3} + 255 \, a^{2} b d^{4} e^{4} + 8840 \, a^{3} d^{3} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{7} e - 408 \, a b^{2} d^{6} e^{2} + 1020 \, a^{2} b d^{5} e^{3} - 1105 \, a^{3} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d} B}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^
6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2*e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*
a*b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a*b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5
+ 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b
^3*d^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/109395*(6435*b^
3*e^8*x^8 + 128*b^3*d^8 - 816*a*b^2*d^7*e + 2040*a^2*b*d^6*e^2 - 2210*a^3*d^5*e^3 + 429*(52*b^3*d*e^7 + 51*a*b
^2*e^8)*x^7 + 33*(802*b^3*d^2*e^6 + 2346*a*b^2*d*e^7 + 765*a^2*b*e^8)*x^6 + 9*(1212*b^3*d^3*e^5 + 10506*a*b^2*
d^2*e^6 + 10200*a^2*b*d*e^7 + 1105*a^3*e^8)*x^5 + 5*(7*b^3*d^4*e^4 + 8160*a*b^2*d^3*e^5 + 23358*a^2*b*d^2*e^6
+ 7514*a^3*d*e^7)*x^4 - 5*(8*b^3*d^5*e^3 - 51*a*b^2*d^4*e^4 - 10812*a^2*b*d^3*e^5 - 10166*a^3*d^2*e^6)*x^3 + 3
*(16*b^3*d^6*e^2 - 102*a*b^2*d^5*e^3 + 255*a^2*b*d^4*e^4 + 8840*a^3*d^3*e^5)*x^2 - (64*b^3*d^7*e - 408*a*b^2*d
^6*e^2 + 1020*a^2*b*d^5*e^3 - 1105*a^3*d^4*e^4)*x)*sqrt(e*x + d)*B/e^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((A + B*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*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)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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