3.13.21 \(\int (A+B x) (d+e x)^{7/2} (b x+c x^2)^2 \, dx\) [1221]

Optimal. Leaf size=267 \[ -\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{9/2}}{9 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{15/2}}{15 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{17/2}}{17 e^6}+\frac {2 B c^2 (d+e x)^{19/2}}{19 e^6} \]

[Out]

-2/9*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(9/2)/e^6+2/11*d*(-b*e+c*d)*(B*d*(-3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(
e*x+d)^(11/2)/e^6+2/13*(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(13/2
)/e^6-2/15*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(15/2)/e^6-2/17*c*(-A*c*e-2*B*b*e+5
*B*c*d)*(e*x+d)^(17/2)/e^6+2/19*B*c^2*(e*x+d)^(19/2)/e^6

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Rubi [A]
time = 0.14, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785} \begin {gather*} -\frac {2 (d+e x)^{15/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{15 e^6}+\frac {2 (d+e x)^{13/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{13 e^6}-\frac {2 d^2 (d+e x)^{9/2} (B d-A e) (c d-b e)^2}{9 e^6}-\frac {2 c (d+e x)^{17/2} (-A c e-2 b B e+5 B c d)}{17 e^6}+\frac {2 d (d+e x)^{11/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{11 e^6}+\frac {2 B c^2 (d+e x)^{19/2}}{19 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(11/2))/(11*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*
d*e + 3*b^2*e^2))*(d + e*x)^(13/2))/(13*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2
))*(d + e*x)^(15/2))/(15*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(17/2))/(17*e^6) + (2*B*c^2*(d + e*
x)^(19/2))/(19*e^6)

Rule 785

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

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{7/2}}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{9/2}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{11/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{13/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{15/2}}{e^5}+\frac {B c^2 (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{9/2}}{9 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{15/2}}{15 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{17/2}}{17 e^6}+\frac {2 B c^2 (d+e x)^{19/2}}{19 e^6}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 273, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (19 A e \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )+B \left (323 b^2 e^2 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+38 b c e \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )-5 c^2 \left (256 d^5-1152 d^4 e x+3168 d^3 e^2 x^2-6864 d^2 e^3 x^3+12870 d e^4 x^4-21879 e^5 x^5\right )\right )\right )}{2078505 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(19*A*e*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 34*b*c*e*(-16*d^3 + 72*d^2*e*x - 198*
d*e^2*x^2 + 429*e^3*x^3) + c^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)) + B
*(323*b^2*e^2*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 38*b*c*e*(128*d^4 - 576*d^3*e*x + 1584*d^
2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) - 5*c^2*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x^2 - 6864*d^2*e^3*x
^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5))))/(2078505*e^6)

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Maple [A]
time = 0.60, size = 278, normalized size = 1.04 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^6*(1/19*B*c^2*(e*x+d)^(19/2)+1/17*((A*e-3*B*d)*c^2+2*B*(b*e-c*d)*c)*(e*x+d)^(17/2)+1/15*((-2*(A*e-B*d)*d+B
*d^2)*c^2+2*(A*e-3*B*d)*(b*e-c*d)*c+B*(b*e-c*d)^2)*(e*x+d)^(15/2)+1/13*((A*e-B*d)*d^2*c^2+2*(-2*(A*e-B*d)*d+B*
d^2)*(b*e-c*d)*c+(A*e-3*B*d)*(b*e-c*d)^2)*(e*x+d)^(13/2)+1/11*(2*(A*e-B*d)*d^2*(b*e-c*d)*c+(-2*(A*e-B*d)*d+B*d
^2)*(b*e-c*d)^2)*(e*x+d)^(11/2)+1/9*(A*e-B*d)*d^2*(b*e-c*d)^2*(e*x+d)^(9/2))

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Maxima [A]
time = 0.28, size = 308, normalized size = 1.15 \begin {gather*} \frac {2}{2078505} \, {\left (109395 \, {\left (x e + d\right )}^{\frac {19}{2}} B c^{2} - 122265 \, {\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} {\left (x e + d\right )}^{\frac {17}{2}} + 138567 \, {\left (10 \, B c^{2} d^{2} + B b^{2} e^{2} + 2 \, A b c e^{2} - 4 \, {\left (2 \, B b c e + A c^{2} e\right )} d\right )} {\left (x e + d\right )}^{\frac {15}{2}} - 159885 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{2} + 3 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 188955 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{3} + 3 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 230945 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c e + A c^{2} e\right )} d^{4} + {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{3}\right )} {\left (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2078505*(109395*(x*e + d)^(19/2)*B*c^2 - 122265*(5*B*c^2*d - 2*B*b*c*e - A*c^2*e)*(x*e + d)^(17/2) + 138567*
(10*B*c^2*d^2 + B*b^2*e^2 + 2*A*b*c*e^2 - 4*(2*B*b*c*e + A*c^2*e)*d)*(x*e + d)^(15/2) - 159885*(10*B*c^2*d^3 -
A*b^2*e^3 - 6*(2*B*b*c*e + A*c^2*e)*d^2 + 3*(B*b^2*e^2 + 2*A*b*c*e^2)*d)*(x*e + d)^(13/2) + 188955*(5*B*c^2*d
^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c*e + A*c^2*e)*d^3 + 3*(B*b^2*e^2 + 2*A*b*c*e^2)*d^2)*(x*e + d)^(11/2) - 230945*
(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c*e + A*c^2*e)*d^4 + (B*b^2*e^2 + 2*A*b*c*e^2)*d^3)*(x*e + d)^(9/2))*e^(-6
)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (254) = 508\).
time = 3.95, size = 545, normalized size = 2.04 \begin {gather*} -\frac {2}{2078505} \, {\left (1280 \, B c^{2} d^{9} - 33 \, {\left (3315 \, B c^{2} x^{9} + 4845 \, A b^{2} x^{6} + 3705 \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + 4199 \, {\left (B b^{2} + 2 \, A b c\right )} x^{7}\right )} e^{9} - 6 \, {\left (62205 \, B c^{2} d x^{8} + 96900 \, A b^{2} d x^{5} + 70642 \, {\left (2 \, B b c + A c^{2}\right )} d x^{7} + 81719 \, {\left (B b^{2} + 2 \, A b c\right )} d x^{6}\right )} e^{8} - 2 \, {\left (216645 \, B c^{2} d^{2} x^{7} + 369835 \, A b^{2} d^{2} x^{4} + 251427 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{6} + 299421 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} x^{5}\right )} e^{7} - 4 \, {\left (43230 \, B c^{2} d^{3} x^{6} + 85595 \, A b^{2} d^{3} x^{3} + 51813 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x^{5} + 64600 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} x^{4}\right )} e^{6} - 5 \, {\left (63 \, B c^{2} d^{4} x^{5} + 969 \, A b^{2} d^{4} x^{2} + 133 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} x^{4} + 323 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} x^{3}\right )} e^{5} + 2 \, {\left (175 \, B c^{2} d^{5} x^{4} + 3230 \, A b^{2} d^{5} x + 380 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} x^{3} + 969 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} x^{2}\right )} e^{4} - 8 \, {\left (50 \, B c^{2} d^{6} x^{3} + 1615 \, A b^{2} d^{6} + 114 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} x^{2} + 323 \, {\left (B b^{2} + 2 \, A b c\right )} d^{6} x\right )} e^{3} + 16 \, {\left (30 \, B c^{2} d^{7} x^{2} + 76 \, {\left (2 \, B b c + A c^{2}\right )} d^{7} x + 323 \, {\left (B b^{2} + 2 \, A b c\right )} d^{7}\right )} e^{2} - 128 \, {\left (5 \, B c^{2} d^{8} x + 19 \, {\left (2 \, B b c + A c^{2}\right )} d^{8}\right )} e\right )} \sqrt {x e + d} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2078505*(1280*B*c^2*d^9 - 33*(3315*B*c^2*x^9 + 4845*A*b^2*x^6 + 3705*(2*B*b*c + A*c^2)*x^8 + 4199*(B*b^2 +
2*A*b*c)*x^7)*e^9 - 6*(62205*B*c^2*d*x^8 + 96900*A*b^2*d*x^5 + 70642*(2*B*b*c + A*c^2)*d*x^7 + 81719*(B*b^2 +
2*A*b*c)*d*x^6)*e^8 - 2*(216645*B*c^2*d^2*x^7 + 369835*A*b^2*d^2*x^4 + 251427*(2*B*b*c + A*c^2)*d^2*x^6 + 2994
21*(B*b^2 + 2*A*b*c)*d^2*x^5)*e^7 - 4*(43230*B*c^2*d^3*x^6 + 85595*A*b^2*d^3*x^3 + 51813*(2*B*b*c + A*c^2)*d^3
*x^5 + 64600*(B*b^2 + 2*A*b*c)*d^3*x^4)*e^6 - 5*(63*B*c^2*d^4*x^5 + 969*A*b^2*d^4*x^2 + 133*(2*B*b*c + A*c^2)*
d^4*x^4 + 323*(B*b^2 + 2*A*b*c)*d^4*x^3)*e^5 + 2*(175*B*c^2*d^5*x^4 + 3230*A*b^2*d^5*x + 380*(2*B*b*c + A*c^2)
*d^5*x^3 + 969*(B*b^2 + 2*A*b*c)*d^5*x^2)*e^4 - 8*(50*B*c^2*d^6*x^3 + 1615*A*b^2*d^6 + 114*(2*B*b*c + A*c^2)*d
^6*x^2 + 323*(B*b^2 + 2*A*b*c)*d^6*x)*e^3 + 16*(30*B*c^2*d^7*x^2 + 76*(2*B*b*c + A*c^2)*d^7*x + 323*(B*b^2 + 2
*A*b*c)*d^7)*e^2 - 128*(5*B*c^2*d^8*x + 19*(2*B*b*c + A*c^2)*d^8)*e)*sqrt(x*e + d)*e^(-6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (272) = 544\).
time = 0.97, size = 1352, normalized size = 5.06 \begin {gather*} \begin {cases} \frac {16 A b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {64 A b c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {32 A b c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {8 A b c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {4 A b c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {640 A b c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {824 A b c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {184 A b c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {4 A b c e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 A c^{2} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 A c^{2} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 A c^{2} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 A c^{2} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 A c^{2} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 A c^{2} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 A c^{2} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 A c^{2} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 A c^{2} e^{3} x^{8} \sqrt {d + e x}}{17} - \frac {32 B b^{2} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B b^{2} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B b^{2} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B b^{2} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B b^{2} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B b^{2} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B b^{2} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B b^{2} e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {512 B b c d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {256 B b c d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {64 B b c d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {32 B b c d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {28 B b c d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {4848 B b c d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {3208 B b c d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {208 B b c d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {4 B b c e^{3} x^{8} \sqrt {d + e x}}{17} - \frac {512 B c^{2} d^{9} \sqrt {d + e x}}{415701 e^{6}} + \frac {256 B c^{2} d^{8} x \sqrt {d + e x}}{415701 e^{5}} - \frac {64 B c^{2} d^{7} x^{2} \sqrt {d + e x}}{138567 e^{4}} + \frac {160 B c^{2} d^{6} x^{3} \sqrt {d + e x}}{415701 e^{3}} - \frac {140 B c^{2} d^{5} x^{4} \sqrt {d + e x}}{415701 e^{2}} + \frac {14 B c^{2} d^{4} x^{5} \sqrt {d + e x}}{46189 e} + \frac {2096 B c^{2} d^{3} x^{6} \sqrt {d + e x}}{12597} + \frac {404 B c^{2} d^{2} e x^{7} \sqrt {d + e x}}{969} + \frac {116 B c^{2} d e^{2} x^{8} \sqrt {d + e x}}{323} + \frac {2 B c^{2} e^{3} x^{9} \sqrt {d + e x}}{19} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {A b^{2} x^{3}}{3} + \frac {A b c x^{4}}{2} + \frac {A c^{2} x^{5}}{5} + \frac {B b^{2} x^{4}}{4} + \frac {2 B b c x^{5}}{5} + \frac {B c^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*A*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*b**2*d**
4*x**2*sqrt(d + e*x)/(429*e) + 424*A*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/
1287 + 80*A*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*A*b*c*d**7*sqrt(d +
e*x)/(6435*e**4) + 32*A*b*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 8*A*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4
*A*b*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 640*A*b*c*d**3*x**4*sqrt(d + e*x)/1287 + 824*A*b*c*d**2*e*x**5*sqrt(
d + e*x)/715 + 184*A*b*c*d*e**2*x**6*sqrt(d + e*x)/195 + 4*A*b*c*e**3*x**7*sqrt(d + e*x)/15 + 256*A*c**2*d**8*
sqrt(d + e*x)/(109395*e**5) - 128*A*c**2*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*A*c**2*d**6*x**2*sqrt(d + e*x
)/(36465*e**3) - 16*A*c**2*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*A*c**2*d**4*x**4*sqrt(d + e*x)/(21879*e)
+ 2424*A*c**2*d**3*x**5*sqrt(d + e*x)/12155 + 1604*A*c**2*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*A*c**2*d*e**2*x
**7*sqrt(d + e*x)/255 + 2*A*c**2*e**3*x**8*sqrt(d + e*x)/17 - 32*B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*
b**2*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt
(d + e*x)/(1287*e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B
*b**2*d*e**2*x**6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15 + 512*B*b*c*d**8*sqrt(d + e*x)/(1093
95*e**5) - 256*B*b*c*d**7*x*sqrt(d + e*x)/(109395*e**4) + 64*B*b*c*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 32*B
*b*c*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 28*B*b*c*d**4*x**4*sqrt(d + e*x)/(21879*e) + 4848*B*b*c*d**3*x**5*
sqrt(d + e*x)/12155 + 3208*B*b*c*d**2*e*x**6*sqrt(d + e*x)/3315 + 208*B*b*c*d*e**2*x**7*sqrt(d + e*x)/255 + 4*
B*b*c*e**3*x**8*sqrt(d + e*x)/17 - 512*B*c**2*d**9*sqrt(d + e*x)/(415701*e**6) + 256*B*c**2*d**8*x*sqrt(d + e*
x)/(415701*e**5) - 64*B*c**2*d**7*x**2*sqrt(d + e*x)/(138567*e**4) + 160*B*c**2*d**6*x**3*sqrt(d + e*x)/(41570
1*e**3) - 140*B*c**2*d**5*x**4*sqrt(d + e*x)/(415701*e**2) + 14*B*c**2*d**4*x**5*sqrt(d + e*x)/(46189*e) + 209
6*B*c**2*d**3*x**6*sqrt(d + e*x)/12597 + 404*B*c**2*d**2*e*x**7*sqrt(d + e*x)/969 + 116*B*c**2*d*e**2*x**8*sqr
t(d + e*x)/323 + 2*B*c**2*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7/2)*(A*b**2*x**3/3 + A*b*c*x**4/2 + A*c
**2*x**5/5 + B*b**2*x**4/4 + 2*B*b*c*x**5/5 + B*c**2*x**6/6), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2710 vs. \(2 (254) = 508\).
time = 1.05, size = 2710, normalized size = 10.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/14549535*(969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^4*e^(-2) + 415701
*(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*b^2*d^4*e^(-3) +
831402*(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*c*d^4*e
^(-3) + 92378*(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*c*d^4*e^(-4) + 46189*(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*c^2*d^4*e^(-4) + 20995*(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*c^2*d^4*e^(-5) + 1662804*(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^2*d^3*e^(-2) + 184756*(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^2*d^3*e^(-3) + 369512*(35*(x*e + d)^(9/2) - 1
80*(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*b*c*d^3*e^
(-3) + 167960*(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*c*d^3*e^(-4) + 83980*(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*c^2*d^3*e^(-4) + 19380*(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*c^
2*d^3*e^(-5) + 277134*(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*b^2*d^2*e^(-2) + 125970*(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^2*d^2*e
^(-3) + 251940*(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*b*c*d^2*e^(-3) + 58140*(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*b*c*d^2*e^(-4) + 29070*(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*c^2*d^2*e^(-4) + 13566*(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 + 150
15*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*c^2*d^2*e^(-5) + 83980*(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*b^2*d*e^(-2) + 19380*(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*b^2*d*e^(
-3) + 38760*(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*c*d*e^(-3) + 18088*(42
9*(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*c*d*
e^(-4) + 9044*(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*c^2*d*e^(-4) + 532*(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*c^2*d*e^(-5) + 4845*(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 - 600
6*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^2*e^(-2) + 2261*(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^2*e^(-3) + 4522*(429*(x*e + d)^(15/2) - 346
5*(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 - 27
027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*A*b*c*e^(-3) + 266*(6435*(x*e +
d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(...

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Mupad [B]
time = 1.62, size = 254, normalized size = 0.95 \begin {gather*} \frac {{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{17\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{15\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{19/2}}{19\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{11\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(17/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(17*e^6) + ((d + e*x)^(13/2)*(2*A*b^2*e^3 - 20*B*c^2*d
^3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(13*e^6) + ((d + e*x)^(15/2)*(2*B*b^2*
e^2 + 20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(15*e^6) + (2*B*c^2*(d + e*x)^(19/2))/(19*e^6)
- (2*d*(b*e - c*d)*(d + e*x)^(11/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(11*e^6) + (2*d^2*(A*e -
B*d)*(b*e - c*d)^2*(d + e*x)^(9/2))/(9*e^6)

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