Optimal. Leaf size=232 \[ \frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}-\frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b} \]
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Rubi [A] time = 0.25, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {50, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}-\frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \sqrt {a+b x} (c+d x)^{3/4} \, dx &=\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}} \, dx}{3 b}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac {\left (2 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{15 b d}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac {\left (8 (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b d^2}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac {\left (8 (b c-a d)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b^{3/2} d^2}-\frac {\left (8 (b c-a d)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b^{3/2} d^2}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac {\left (8 (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b^{3/2} d^2 \sqrt {a+b x}}-\frac {\left (8 (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b^{3/2} d^2 \sqrt {a+b x}}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}-\frac {\left (8 (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{15 b^{3/2} d^2 \sqrt {a+b x}}\\ &=\frac {4 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 b d}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}+\frac {8 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.31 \[ \frac {2 (a+b x)^{3/2} (c+d x)^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {5}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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