3.21.43 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2) \, dx\) [2043]

Optimal. Leaf size=100 \[ -\frac {2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^4}+\frac {2 b (b d-a e)^2 (d+e x)^{9/2}}{3 e^4}-\frac {6 b^2 (b d-a e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^3 (d+e x)^{13/2}}{13 e^4} \]

[Out]

-2/7*(-a*e+b*d)^3*(e*x+d)^(7/2)/e^4+2/3*b*(-a*e+b*d)^2*(e*x+d)^(9/2)/e^4-6/11*b^2*(-a*e+b*d)*(e*x+d)^(11/2)/e^
4+2/13*b^3*(e*x+d)^(13/2)/e^4

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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45} \begin {gather*} -\frac {6 b^2 (d+e x)^{11/2} (b d-a e)}{11 e^4}+\frac {2 b (d+e x)^{9/2} (b d-a e)^2}{3 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^3}{7 e^4}+\frac {2 b^3 (d+e x)^{13/2}}{13 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(-2*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^4) + (2*b*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^4) - (6*b^2*(b*d - a*e)*
(d + e*x)^(11/2))/(11*e^4) + (2*b^3*(d + e*x)^(13/2))/(13*e^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 102, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(429*a^3*e^3 + 143*a^2*b*e^2*(-2*d + 7*e*x) + 13*a*b^2*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) +
b^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3)))/(3003*e^4)

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Maple [A]
time = 0.15, size = 147, normalized size = 1.47

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 b^{3} e^{3} x^{3}+819 a \,b^{2} e^{3} x^{2}-126 b^{3} d \,e^{2} x^{2}+1001 a^{2} b \,e^{3} x -364 a \,b^{2} d \,e^{2} x +56 b^{3} d^{2} e x +429 a^{3} e^{3}-286 a^{2} b d \,e^{2}+104 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{3003 e^{4}}\)	\(116\)
derivativedivides	\(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\)	\(147\)
default	\(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\)	\(147\)
trager	\(\frac {2 \left (231 b^{3} e^{6} x^{6}+819 a \,b^{2} e^{6} x^{5}+567 b^{3} d \,e^{5} x^{5}+1001 a^{2} b \,e^{6} x^{4}+2093 a \,b^{2} d \,e^{5} x^{4}+371 b^{3} d^{2} e^{4} x^{4}+429 a^{3} e^{6} x^{3}+2717 a^{2} b d \,e^{5} x^{3}+1469 a \,b^{2} d^{2} e^{4} x^{3}+5 b^{3} d^{3} e^{3} x^{3}+1287 a^{3} d \,e^{5} x^{2}+2145 a^{2} b \,d^{2} e^{4} x^{2}+39 a \,b^{2} d^{3} e^{3} x^{2}-6 b^{3} d^{4} e^{2} x^{2}+1287 a^{3} d^{2} e^{4} x +143 a^{2} b \,d^{3} e^{3} x -52 a \,b^{2} d^{4} e^{2} x +8 b^{3} d^{5} e x +429 a^{3} d^{3} e^{3}-286 a^{2} b \,d^{4} e^{2}+104 a \,b^{2} d^{5} e -16 b^{3} d^{6}\right ) \sqrt {e x +d}}{3003 e^{4}}\)	\(286\)
risch	\(\frac {2 \left (231 b^{3} e^{6} x^{6}+819 a \,b^{2} e^{6} x^{5}+567 b^{3} d \,e^{5} x^{5}+1001 a^{2} b \,e^{6} x^{4}+2093 a \,b^{2} d \,e^{5} x^{4}+371 b^{3} d^{2} e^{4} x^{4}+429 a^{3} e^{6} x^{3}+2717 a^{2} b d \,e^{5} x^{3}+1469 a \,b^{2} d^{2} e^{4} x^{3}+5 b^{3} d^{3} e^{3} x^{3}+1287 a^{3} d \,e^{5} x^{2}+2145 a^{2} b \,d^{2} e^{4} x^{2}+39 a \,b^{2} d^{3} e^{3} x^{2}-6 b^{3} d^{4} e^{2} x^{2}+1287 a^{3} d^{2} e^{4} x +143 a^{2} b \,d^{3} e^{3} x -52 a \,b^{2} d^{4} e^{2} x +8 b^{3} d^{5} e x +429 a^{3} d^{3} e^{3}-286 a^{2} b \,d^{4} e^{2}+104 a \,b^{2} d^{5} e -16 b^{3} d^{6}\right ) \sqrt {e x +d}}{3003 e^{4}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/13*b^3*(e*x+d)^(13/2)+1/11*((a*e-b*d)*b^2+b*(2*a*b*e-2*b^2*d))*(e*x+d)^(11/2)+1/9*((a*e-b*d)*(2*a*b*e
-2*b^2*d)+b*(a^2*e^2-2*a*b*d*e+b^2*d^2))*(e*x+d)^(9/2)+1/7*(a*e-b*d)*(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)
)

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Maxima [A]
time = 0.29, size = 121, normalized size = 1.21 \begin {gather*} \frac {2}{3003} \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} d - a b^{2} e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/3003*(231*(x*e + d)^(13/2)*b^3 - 819*(b^3*d - a*b^2*e)*(x*e + d)^(11/2) + 1001*(b^3*d^2 - 2*a*b^2*d*e + a^2*
b*e^2)*(x*e + d)^(9/2) - 429*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(x*e + d)^(7/2))*e^(-4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (87) = 174\).
time = 2.27, size = 255, normalized size = 2.55 \begin {gather*} -\frac {2}{3003} \, {\left (16 \, b^{3} d^{6} - {\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} e^{6} - {\left (567 \, b^{3} d x^{5} + 2093 \, a b^{2} d x^{4} + 2717 \, a^{2} b d x^{3} + 1287 \, a^{3} d x^{2}\right )} e^{5} - {\left (371 \, b^{3} d^{2} x^{4} + 1469 \, a b^{2} d^{2} x^{3} + 2145 \, a^{2} b d^{2} x^{2} + 1287 \, a^{3} d^{2} x\right )} e^{4} - {\left (5 \, b^{3} d^{3} x^{3} + 39 \, a b^{2} d^{3} x^{2} + 143 \, a^{2} b d^{3} x + 429 \, a^{3} d^{3}\right )} e^{3} + 2 \, {\left (3 \, b^{3} d^{4} x^{2} + 26 \, a b^{2} d^{4} x + 143 \, a^{2} b d^{4}\right )} e^{2} - 8 \, {\left (b^{3} d^{5} x + 13 \, a b^{2} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-2/3003*(16*b^3*d^6 - (231*b^3*x^6 + 819*a*b^2*x^5 + 1001*a^2*b*x^4 + 429*a^3*x^3)*e^6 - (567*b^3*d*x^5 + 2093
*a*b^2*d*x^4 + 2717*a^2*b*d*x^3 + 1287*a^3*d*x^2)*e^5 - (371*b^3*d^2*x^4 + 1469*a*b^2*d^2*x^3 + 2145*a^2*b*d^2
*x^2 + 1287*a^3*d^2*x)*e^4 - (5*b^3*d^3*x^3 + 39*a*b^2*d^3*x^2 + 143*a^2*b*d^3*x + 429*a^3*d^3)*e^3 + 2*(3*b^3
*d^4*x^2 + 26*a*b^2*d^4*x + 143*a^2*b*d^4)*e^2 - 8*(b^3*d^5*x + 13*a*b^2*d^5)*e)*sqrt(x*e + d)*e^(-4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (92) = 184\).
time = 0.52, size = 549, normalized size = 5.49 \begin {gather*} \begin {cases} \frac {2 a^{3} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a^{3} d^{2} x \sqrt {d + e x}}{7} + \frac {6 a^{3} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a^{3} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {4 a^{2} b d^{4} \sqrt {d + e x}}{21 e^{2}} + \frac {2 a^{2} b d^{3} x \sqrt {d + e x}}{21 e} + \frac {10 a^{2} b d^{2} x^{2} \sqrt {d + e x}}{7} + \frac {38 a^{2} b d e x^{3} \sqrt {d + e x}}{21} + \frac {2 a^{2} b e^{2} x^{4} \sqrt {d + e x}}{3} + \frac {16 a b^{2} d^{5} \sqrt {d + e x}}{231 e^{3}} - \frac {8 a b^{2} d^{4} x \sqrt {d + e x}}{231 e^{2}} + \frac {2 a b^{2} d^{3} x^{2} \sqrt {d + e x}}{77 e} + \frac {226 a b^{2} d^{2} x^{3} \sqrt {d + e x}}{231} + \frac {46 a b^{2} d e x^{4} \sqrt {d + e x}}{33} + \frac {6 a b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 b^{3} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 b^{3} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 b^{3} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 b^{3} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 b^{3} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 b^{3} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{3} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\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)**(5/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**3*d**3*sqrt(d + e*x)/(7*e) + 6*a**3*d**2*x*sqrt(d + e*x)/7 + 6*a**3*d*e*x**2*sqrt(d + e*x)/7 +
2*a**3*e**2*x**3*sqrt(d + e*x)/7 - 4*a**2*b*d**4*sqrt(d + e*x)/(21*e**2) + 2*a**2*b*d**3*x*sqrt(d + e*x)/(21*
e) + 10*a**2*b*d**2*x**2*sqrt(d + e*x)/7 + 38*a**2*b*d*e*x**3*sqrt(d + e*x)/21 + 2*a**2*b*e**2*x**4*sqrt(d + e
*x)/3 + 16*a*b**2*d**5*sqrt(d + e*x)/(231*e**3) - 8*a*b**2*d**4*x*sqrt(d + e*x)/(231*e**2) + 2*a*b**2*d**3*x**
2*sqrt(d + e*x)/(77*e) + 226*a*b**2*d**2*x**3*sqrt(d + e*x)/231 + 46*a*b**2*d*e*x**4*sqrt(d + e*x)/33 + 6*a*b*
*2*e**2*x**5*sqrt(d + e*x)/11 - 32*b**3*d**6*sqrt(d + e*x)/(3003*e**4) + 16*b**3*d**5*x*sqrt(d + e*x)/(3003*e*
*3) - 4*b**3*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*b**3*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*b**3*d**2*x*
*4*sqrt(d + e*x)/429 + 54*b**3*d*e*x**5*sqrt(d + e*x)/143 + 2*b**3*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**
(5/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (87) = 174\).
time = 1.11, size = 908, normalized size = 9.08 \begin {gather*} \frac {2}{15015} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d^{3} e^{\left (-1\right )} + 3003 \, {\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 b^{2} d^{3} e^{\left (-2\right )} + 429 \, {\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^{3} d^{3} e^{\left (-3\right )} + 9009 \, {\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^{2} b d^{2} e^{\left (-1\right )} + 3861 \, {\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^{2} d^{2} e^{\left (-2\right )} + 143 \, {\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^{3} d^{2} e^{\left (-3\right )} + 15015 \, \sqrt {x e + d} a^{3} d^{3} + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d^{2} + 3861 \, {\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^{2} b d e^{\left (-1\right )} + 429 \, {\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 )} a b^{2} d e^{\left (-2\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{3} d e^{\left (-3\right )} + 3003 \, {\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^{3} d + 143 \, {\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 )} a^{2} b e^{\left (-1\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{2} e^{\left (-2\right )} + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{3} e^{\left (-3\right )} + 429 \, {\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^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/15015*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^3*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)
^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^3*e^(-2) + 429*(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^3*d^3*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt
(x*e + d)*d^2)*a^2*b*d^2*e^(-1) + 3861*(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^2*e^(-2) + 143*(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^3*d^2*e^(-3) + 15015*sqrt(x*e + d)*a^3*d^3 + 15015*((x
*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d^2 + 3861*(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^2*b*d*e^(-1) + 429*(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*e^(-2) + 65*(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^3*d*e^(-3) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*d + 143
*(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^2*b*e^(-1) + 65*(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*e^(-2) + 5*(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 - 60
06*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^3*e^(-3) + 429*(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^3)*e^(-1)

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Mupad [B]
time = 2.07, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

(2*b^3*(d + e*x)^(13/2))/(13*e^4) - ((6*b^3*d - 6*a*b^2*e)*(d + e*x)^(11/2))/(11*e^4) + (2*(a*e - b*d)^3*(d +
e*x)^(7/2))/(7*e^4) + (2*b*(a*e - b*d)^2*(d + e*x)^(9/2))/(3*e^4)

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