3.4.23 \(\int (d+e x)^{7/2} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=147 \[ \frac {2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac {4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac {4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac {4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac {4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(15/2))/(15*e^5)
+ (2*c^2*(d + e*x)^(17/2))/(17*e^5)

Rule 698

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 124, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{9/2} \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 )}{109395 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(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)))/(109395*e^
5)

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IntegrateAlgebraic [A]  time = 0.08, size = 164, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (12155 b^2 d^2 e^2-19890 b^2 d e^2 (d+e x)+8415 b^2 e^2 (d+e x)^2-24310 b c d^3 e+59670 b c d^2 e (d+e x)-50490 b c d e (d+e x)^2+14586 b c e (d+e x)^3+12155 c^2 d^4-39780 c^2 d^3 (d+e x)+50490 c^2 d^2 (d+e x)^2-29172 c^2 d (d+e x)^3+6435 c^2 (d+e x)^4\right )}{109395 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(12155*c^2*d^4 - 24310*b*c*d^3*e + 12155*b^2*d^2*e^2 - 39780*c^2*d^3*(d + e*x) + 59670*b*c*
d^2*e*(d + e*x) - 19890*b^2*d*e^2*(d + e*x) + 50490*c^2*d^2*(d + e*x)^2 - 50490*b*c*d*e*(d + e*x)^2 + 8415*b^2
*e^2*(d + e*x)^2 - 29172*c^2*d*(d + e*x)^3 + 14586*b*c*e*(d + e*x)^3 + 6435*c^2*(d + e*x)^4))/(109395*e^5)

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fricas [B]  time = 0.42, size = 290, normalized size = 1.97 \begin {gather*} \frac {2 \, {\left (6435 \, c^{2} e^{8} x^{8} + 128 \, c^{2} d^{8} - 544 \, b c d^{7} e + 680 \, b^{2} d^{6} e^{2} + 858 \, {\left (26 \, c^{2} d e^{7} + 17 \, b c e^{8}\right )} x^{7} + 33 \, {\left (802 \, c^{2} d^{2} e^{6} + 1564 \, b c d e^{7} + 255 \, b^{2} e^{8}\right )} x^{6} + 36 \, {\left (303 \, c^{2} d^{3} e^{5} + 1751 \, b c d^{2} e^{6} + 850 \, b^{2} d e^{7}\right )} x^{5} + 5 \, {\left (7 \, c^{2} d^{4} e^{4} + 5440 \, b c d^{3} e^{5} + 7786 \, b^{2} d^{2} e^{6}\right )} x^{4} - 10 \, {\left (4 \, c^{2} d^{5} e^{3} - 17 \, b c d^{4} e^{4} - 1802 \, b^{2} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{6} e^{2} - 68 \, b c d^{5} e^{3} + 85 \, b^{2} d^{4} e^{4}\right )} x^{2} - 4 \, {\left (16 \, c^{2} d^{7} e - 68 \, b c d^{6} e^{2} + 85 \, b^{2} d^{5} e^{3}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*(6435*c^2*e^8*x^8 + 128*c^2*d^8 - 544*b*c*d^7*e + 680*b^2*d^6*e^2 + 858*(26*c^2*d*e^7 + 17*b*c*e^8)*x
^7 + 33*(802*c^2*d^2*e^6 + 1564*b*c*d*e^7 + 255*b^2*e^8)*x^6 + 36*(303*c^2*d^3*e^5 + 1751*b*c*d^2*e^6 + 850*b^
2*d*e^7)*x^5 + 5*(7*c^2*d^4*e^4 + 5440*b*c*d^3*e^5 + 7786*b^2*d^2*e^6)*x^4 - 10*(4*c^2*d^5*e^3 - 17*b*c*d^4*e^
4 - 1802*b^2*d^3*e^5)*x^3 + 3*(16*c^2*d^6*e^2 - 68*b*c*d^5*e^3 + 85*b^2*d^4*e^4)*x^2 - 4*(16*c^2*d^7*e - 68*b*
c*d^6*e^2 + 85*b^2*d^5*e^3)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.25, size = 1245, normalized size = 8.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d^4*e^(-2) + 43758*(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*c*d^4*e^(-3) + 2431*(3
5*(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)*c^2*d^4*e^(-4) + 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)*b^2*d^3*e^(-2) + 19448*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b*c*d^3*e^(-3) + 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)*c
^2*d^3*e^(-4) + 14586*(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^2*d^2*e^(-2) + 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)*b*c*d^2*e^(-3)
+ 1530*(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)*c^2*d^2*e^(-4) + 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^2*d*e^(-2) + 2040*(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*d*e^(-3) + 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)*c^2*d*e^(-4) + 255*(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^2*e^(-2) + 238*(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 - 643
5*sqrt(x*e + d)*d^7)*b*c*e^(-3) + 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)*c^2*e^(-4))*e^(-1)

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maple [A]  time = 0.05, size = 141, normalized size = 0.96 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 c^{2} x^{4} e^{4}+14586 b c \,e^{4} x^{3}-3432 c^{2} d \,e^{3} x^{3}+8415 b^{2} e^{4} x^{2}-6732 b c d \,e^{3} x^{2}+1584 c^{2} d^{2} e^{2} x^{2}-3060 b^{2} d \,e^{3} x +2448 b c \,d^{2} e^{2} x -576 c^{2} d^{3} e x +680 b^{2} d^{2} e^{2}-544 b c \,d^{3} e +128 c^{2} d^{4}\right )}{109395 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*(e*x+d)^(9/2)*(6435*c^2*e^4*x^4+14586*b*c*e^4*x^3-3432*c^2*d*e^3*x^3+8415*b^2*e^4*x^2-6732*b*c*d*e^3*
x^2+1584*c^2*d^2*e^2*x^2-3060*b^2*d*e^3*x+2448*b*c*d^2*e^2*x-576*c^2*d^3*e*x+680*b^2*d^2*e^2-544*b*c*d^3*e+128
*c^2*d^4)/e^5

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maxima [A]  time = 1.42, size = 139, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{2} - 14586 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 8415 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 19890 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 12155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{109395 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*(6435*(e*x + d)^(17/2)*c^2 - 14586*(2*c^2*d - b*c*e)*(e*x + d)^(15/2) + 8415*(6*c^2*d^2 - 6*b*c*d*e +
b^2*e^2)*(e*x + d)^(13/2) - 19890*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(11/2) + 12155*(c^2*d^4 - 2
*b*c*d^3*e + b^2*d^2*e^2)*(e*x + d)^(9/2))/e^5

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mupad [B]  time = 0.22, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}-\frac {{\left (d+e\,x\right )}^{11/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{13\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(d + e*x)^(17/2))/(17*e^5) - ((d + e*x)^(11/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/(11*e^5) + ((d
+ e*x)^(13/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(13*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(15/2))/(15*e^
5) + (2*d^2*(b*e - c*d)^2*(d + e*x)^(9/2))/(9*e^5)

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sympy [A]  time = 13.61, size = 590, normalized size = 4.01 \begin {gather*} \begin {cases} \frac {16 b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {64 b c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {32 b c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {8 b c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {4 b c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {640 b c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {824 b c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {184 b c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {4 b c e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 c^{2} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 c^{2} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 c^{2} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 c^{2} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 c^{2} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 c^{2} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 c^{2} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 c^{2} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 c^{2} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*b**2*d**4*x**2
*sqrt(d + e*x)/(429*e) + 424*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*
b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*b*c*d**7*sqrt(d + e*x)/(6435*e**4)
+ 32*b*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 8*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4*b*c*d**4*x**3*sqrt(
d + e*x)/(1287*e) + 640*b*c*d**3*x**4*sqrt(d + e*x)/1287 + 824*b*c*d**2*e*x**5*sqrt(d + e*x)/715 + 184*b*c*d*e
**2*x**6*sqrt(d + e*x)/195 + 4*b*c*e**3*x**7*sqrt(d + e*x)/15 + 256*c**2*d**8*sqrt(d + e*x)/(109395*e**5) - 12
8*c**2*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*c**2*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*c**2*d**5*x**3*s
qrt(d + e*x)/(21879*e**2) + 14*c**2*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*c**2*d**3*x**5*sqrt(d + e*x)/1215
5 + 1604*c**2*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*c**2*d*e**2*x**7*sqrt(d + e*x)/255 + 2*c**2*e**3*x**8*sqrt(
d + e*x)/17, Ne(e, 0)), (d**(7/2)*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), True))

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