3.19.87 \(\int (a+b x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=376 \[ \frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)}+\frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (12*b*(b*d - a*e)^5*(d + e
*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
9*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) -
(12*b^5*(b*d - a*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(15/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

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])

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) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 \sqrt {d+e x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 \sqrt {d+e x} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6 \sqrt {d+e x}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac {b^6 (d+e x)^{13/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 163, normalized size = 0.43 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (-20790 b^5 (d+e x)^5 (b d-a e)+61425 b^4 (d+e x)^4 (b d-a e)^2-100100 b^3 (d+e x)^3 (b d-a e)^3+96525 b^2 (d+e x)^2 (b d-a e)^4-54054 b (d+e x) (b d-a e)^5+15015 (b d-a e)^6+3003 b^6 (d+e x)^6\right )}{45045 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(15015*(b*d - a*e)^6 - 54054*b*(b*d - a*e)^5*(d + e*x) + 96525*b^2*(b*d -
a*e)^4*(d + e*x)^2 - 100100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 61425*b^4*(b*d - a*e)^2*(d + e*x)^4 - 20790*b^5*(
b*d - a*e)*(d + e*x)^5 + 3003*b^6*(d + e*x)^6))/(45045*e^7*(a + b*x))

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IntegrateAlgebraic [A]  time = 49.74, size = 466, normalized size = 1.24 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (15015 a^6 e^6+54054 a^5 b e^5 (d+e x)-90090 a^5 b d e^5+225225 a^4 b^2 d^2 e^4+96525 a^4 b^2 e^4 (d+e x)^2-270270 a^4 b^2 d e^4 (d+e x)-300300 a^3 b^3 d^3 e^3+540540 a^3 b^3 d^2 e^3 (d+e x)+100100 a^3 b^3 e^3 (d+e x)^3-386100 a^3 b^3 d e^3 (d+e x)^2+225225 a^2 b^4 d^4 e^2-540540 a^2 b^4 d^3 e^2 (d+e x)+579150 a^2 b^4 d^2 e^2 (d+e x)^2+61425 a^2 b^4 e^2 (d+e x)^4-300300 a^2 b^4 d e^2 (d+e x)^3-90090 a b^5 d^5 e+270270 a b^5 d^4 e (d+e x)-386100 a b^5 d^3 e (d+e x)^2+300300 a b^5 d^2 e (d+e x)^3+20790 a b^5 e (d+e x)^5-122850 a b^5 d e (d+e x)^4+15015 b^6 d^6-54054 b^6 d^5 (d+e x)+96525 b^6 d^4 (d+e x)^2-100100 b^6 d^3 (d+e x)^3+61425 b^6 d^2 (d+e x)^4+3003 b^6 (d+e x)^6-20790 b^6 d (d+e x)^5\right )}{45045 e^6 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(15015*b^6*d^6 - 90090*a*b^5*d^5*e + 225225*a^2*b^4*d^4*e^2 - 300
300*a^3*b^3*d^3*e^3 + 225225*a^4*b^2*d^2*e^4 - 90090*a^5*b*d*e^5 + 15015*a^6*e^6 - 54054*b^6*d^5*(d + e*x) + 2
70270*a*b^5*d^4*e*(d + e*x) - 540540*a^2*b^4*d^3*e^2*(d + e*x) + 540540*a^3*b^3*d^2*e^3*(d + e*x) - 270270*a^4
*b^2*d*e^4*(d + e*x) + 54054*a^5*b*e^5*(d + e*x) + 96525*b^6*d^4*(d + e*x)^2 - 386100*a*b^5*d^3*e*(d + e*x)^2
+ 579150*a^2*b^4*d^2*e^2*(d + e*x)^2 - 386100*a^3*b^3*d*e^3*(d + e*x)^2 + 96525*a^4*b^2*e^4*(d + e*x)^2 - 1001
00*b^6*d^3*(d + e*x)^3 + 300300*a*b^5*d^2*e*(d + e*x)^3 - 300300*a^2*b^4*d*e^2*(d + e*x)^3 + 100100*a^3*b^3*e^
3*(d + e*x)^3 + 61425*b^6*d^2*(d + e*x)^4 - 122850*a*b^5*d*e*(d + e*x)^4 + 61425*a^2*b^4*e^2*(d + e*x)^4 - 207
90*b^6*d*(d + e*x)^5 + 20790*a*b^5*e*(d + e*x)^5 + 3003*b^6*(d + e*x)^6))/(45045*e^6*(a*e + b*e*x))

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fricas [A]  time = 0.44, size = 447, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \, {\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} - {\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^4*d^5*e^2 - 45760*a^3*b^3*d^4*e^3 +
51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^2*e^5 + 15015*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6
*d^2*e^5 - 30*a*b^5*d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a*b^5*d^2*e^5 + 195*a^2*b^4*d*e^6 +
2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^3 - 480*a*b^5*d^3*e^4 + 1560*a^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19
305*a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d^3*e^4 - 5720*a^3*b^3*d^2*e^5 +
6435*a^4*b^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^2*b^4*d^4*e^3 - 2288
0*a^3*b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^6 - 15015*a^6*e^7)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.27, size = 970, normalized size = 2.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(90090*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d*e^(-1)*sgn(b*x + a) + 45045*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d*e^(-2)*sgn(b*x + a) + 25740*(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*b^3*d*e^(-3)*sgn(b*x + a) + 2145*(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^4*d*e^(-4)*sgn(b*x + a) + 390*(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^5*d*e^(-5)*sgn(b*x + a) + 15
*(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^6*d*e^(-6)*sgn(b*x + a) + 18018*(3*
(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b*e^(-1)*sgn(b*x + a) + 19305*(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^4*b^2*e^(-2)*sgn(b*x + a) + 28
60*(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^3*b^3*e^(-3)*sgn(b*x + a) + 975*(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^2*b^4*e^(-4)*sgn(b*x
+ a) + 90*(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^5*e^(-5)*sgn(b*x + a) +
7*(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^
6*e^(-6)*sgn(b*x + a) + 45045*sqrt(x*e + d)*a^6*d*sgn(b*x + a) + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a
^6*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.06, size = 393, normalized size = 1.05 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 a^{6} e^{6}-36036 a^{5} b d \,e^{5}+51480 a^{4} b^{2} d^{2} e^{4}-45760 a^{3} b^{3} d^{3} e^{3}+24960 a^{2} b^{4} d^{4} e^{2}-7680 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+61425*a^2*b^4*e^6*x^4-18900*a*b
^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*
b^6*d^3*e^3*x^3+96525*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*d^3*e^3*x^
2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x
+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^
3*e^3+24960*a^2*b^4*d^4*e^2-7680*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.95, size = 760, normalized size = 2.02 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d} a}{9009 \, e^{6}} + \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} b}{45045 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)*a/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 22880*a
^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^2*e
^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3
*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1430*a^3*b^2*d*e^6 - 6435*a^4*b
*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*
d*e^6 + 3003*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b^2*d^3*e^4
+ 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(e*x + d)*b/e^7

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((a + b*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Integral((a + b*x)*sqrt(d + e*x)*((a + b*x)**2)**(5/2), x)

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